While I work on France, I've decided to put up a list of interesting minor powers for people to pick. Of course, you can pick a country not on this list, but I've tried to point out the more interesting parts of the world (and put into canon more officially a lot of offhand remarks I've made when people ask me about Country X). A map, for reference:
North America
Canada: The nation is locked in a battle over becoming a republic as well as an increasingly militant Quebecois separatist movement. Historically isolationist, Canadians are choosing between increased integration with Britain (for the conservatives) and the USA (for the liberals.)
Mexico: Normally a fairly calm place, the recent landslide election of the
Partido Anarquista Nacional has many observers in Washington on edge as the government experiments in radically free-market economics. They swear they aren't Randists, and don't believe in enacting a "despotism of liberty" to enact radical change, but the US is still suspicious they're a front for the Antioquian drug lords and disgruntled libertists.
Cuba: With no American embargo, Castro was able to develop Cuba a great deal more, and Cuba and Zanzibar duel it out for "party capital of the world."
Central American states: The Partido Americano, a fascist guerrilla group that wants to re-establish the United Provinces of Central America along fascist lines, continues to terrorize the governments from Guatemala to Costa Rica.
South America
Venezuela: The People's Republic of Venezuela is one of the few major oil producing countries that is specifically in the Comintern, which was more important in the Stalinist era than it is today. Not democratic, but pretty relaxed.
Colombia: No matter how much the US prods them, the Colombian government can't seem to crush the Antioquian rebels - probably because Escobar has most of the government ministers in his pocket.
Antioquia: In 1958, a group of American and British radical capitalists inspired by the Objectivist ideologue Ayn Rand received funding from her corporation, Taggart Industries, to filibuster the teetering Colombian government and turn it into a libertarian utopia. The filibusterers made connections with drug lords in Antioquia province and managed to bribe and assassinate enough government officials to proclaim the "Antioquian Free State." The arrangement lasted for about five years, before the drug lords managed to co-opt the Liberty Party and turn the country into a narcostate. In Antioquia, you can buy and sell anything, as long as you don't cross President Escobar. Drug-addled biological terrors skulk through the ruins of Rapture, the now-abandoned purpose-built capital of the Free State. Fighting between Escobar's Liberty Party government, Randist hardliners, Colombian incursions and rival drug lords make it a chaotic place, but somehow business still gets done.
Peru: The People's Republic of Peru was, for a while, the People's Empire of Peru when its wacky dictator decided he was the rightful Emperor of the Spanish Empire, King of Kings, and Jesus's brother. When he invaded Bolivia, a joint Russo-American operation assassinated him and put a government more loyal to Moscow in place. To this day, though, weird cults still exist in the Andes revolving around trying to summon him back to the realm of the living, which will also herald the return of legendary founder-king Manco Capac.
Chile: An eternal battleground between UN and Axis diplomats and spies, Chile has bounced between democracy, left-wing dictatorship, right-wing dictatorship, communist dictatorship, anarcho-capitalist dictatorship (for a few months), and back to democracy again. The current government, led by the social liberal Radical Party, is appealing to the UN for help in putting down a Randist insurrection in Patagonia. Still pretty much first world, despite all that.
Argentina: With a powerful, Integralist Brazil on their doorstep, Argentina was initially a solid member of the UN until a Randist coup overthrew the government in 1953. The new Argentine president, Pedro Eugenio Aramburu, helped codify many of Randism's finer ideological points - the need of a transitory stage of dictatorship to ensure economic and social reform, called the "despotism of liberty," was chief amongst them. His authoritarian radical capitalism was unloved by either the UN or the Axis, and even Rand herself criticized his regime's purging of the Senate. German spies instigated a counter-coup in 1956, bringing Juan Peron back to power in exchange for carefully modifying his platform to be more explicitly fascist. He would then invade the Falkland Islands and force first the British and then the Americans to withdraw in disgrace in a war that lasted until 1973. Argentina remains under a Peronist government today, and is a growing economic power with designs on ursurping Brazil's place as the greatest power in South America.
Europe:
Ireland: Prosperous and the least militarized country in Free Europe, Ireland has historically been France's major partner in furthering European integration. Conveniently, it's also a great way to thumb their noses at the British. Its Sinn Fein government looks set for re-election, which may make the Northern Ireland peace process more difficult still.
Belgium / Luxembourg / Netherlands: The Low Countries are all firm members of the European Cooperation Organization and well-integrated into the French economy. The Dutch have recently put a conservative government in power that may make efforts to resist ECO integration, but for the most part, when France says "frog," the Benelux jumps. Belgium is a major financial center and has a lot of economic muscle, but the Progressive Party resents the dominance of the National Democratic Party and the Belgian Social Union is unhappy with the dominance of Savarin Industries.
Iceland: Loyal British allies in the Council of Europe and European Parliament, Iceland is home to many British and American military bases and agitates for the liberation of its Nordic bretheren in the UNGA. Norway has proven receptive, but the Axis will never let it break its neutrality.
Portugal: After being forced to let go of Angola and Mozambique, and watching the South Africans prop up the remnants of their colonial governments, the Portuguese junta has liberalized quite a bit and even has semi-competitive elections in its national primaries despite being a one-party state. With Spain weak, the UN is wondering if it could stomach Portugal in the ECO and maybe even the UN generally (though it's doubtful Germany would allow the latter.)
Denmark: Probably the most humane of Germany's direct puppet states, the Danes at least get to control their economy, if not their politics. The Danish National Party takes its orders directly from Berlin, despite what its own propaganda says, and the King of Denmark is still in exile in London.
Norway: The only actual democracy in the Axis, Norway has recently be putting out feelers to join the ECO. Berlin would never allow it, however.
Finland: A sort of quasi-democracy, with liberal, socialist and communist parties banned but otherwise with a smorgasboard of choices ranging from Christian democratic to I-can't-believe-it's-not-National Socialism. A solid middle power, and indebted to the Axis for their help in defending against the Soviet Union.
Sweden: A state on the Finnish model, Sweden has far less freedom to act independently from Germany.
Poland: The Polish National Republic is called "the armpit of Europe," and is more a reservation for unwanted Poles (i.e. all of them) in Germany than a functioning state. It is totally and utterly dominated by German political and economic interests and is only really theoretically independent. It's dirt poor, too - worse than many countries in sub-Saharan Africa.
Baltic Union: Being "German-ish" and having once been directly integrated into Grossdeutschland has given the Balts something of a pass compared to the other German puppets in the region. They may only produce cheap industrial goods for the Germans, but they at least have their own army and government, and Berlin has fortunately not rolled in the tanks too many times.
Ruthenia: One step above Poland in the "awfulness of buffer state" rankings, the country also known as Belarus is more like a giant prison camp for Slavs than a real country. It's a nasty place, and every day families risk their lives to cross the border into the relative freedom of Ukraine. (The border with the USSR is far too heavily fortified to make escape there practical.)
Ukraine: The Germans received help from some local Ukrainian elements during their war against the USSR, and rewarded them with a fairly autonomous state. In return, Kiev is very loyal to Berlin, and as far as Germany is concerned, Ukrainians are the only good Slavs there are.
Hungary: The only real Central / Eastern European state that could honestly be called an Axis ally rather than a German or Italian puppet, Hungary was the big winner in the Romanian Spring of 1971. They took the rest of Transylvania that the Nazis hadn't already awarded them.
Romania: After trying to implement "nationalism with a human face" in 1971, Romania has been thoroughly chastized, and now hopes that the big mean countries around them will just go away. A faithful Axis servant, but no longer a willing one.
Croatia: The Italian answer to Poland, but not quite as hideous. At this point, most Croatian nationalists are happy they got left with anything at all.
Bosnia: The Italians wanted to spin off the Mediterranean Muslim states so they could focus on assimilating Libya. For whatever reason, Victor Emmanuel III didn't want the Bosnian throne, so it was given to Marshal Pietro Bagdolio. It became a "republic" when his son became Duce in 1980 and had to renounce the title.
Bulgaria: The Germans decided that the Tsar of Bulgaria would be useful in keeping the country in line, and the Tsardom of Bulgaria is a loyal (if marginally independent) German ally.
Greece: A civil war in Greece between Italian-aligned monarchist fascists in government and republican fascists who wanted to pursue neutrality eventually resulted in a republican victory and the loss of the Italian grip on Greece (and, say some, the start of the Italian decline.) The Greco-Turkish "Little Alliance" actually wielded some influence in the region during the 60s, but a series of coups ended Greek relevance. Athens is a city of spies, and every Great Power has their own man in the junta who they hope will become the next President-General.
Turkey: A democracy surrounded by fascist powers, the Turks are cautiously neutral, focusing on internal economic development.
Middle East / North Africa:
Israel: A democracy heavily protected by the UN, the Israelis have a formidable military of their own. While they hate their neighbors only slightly less than they hate them (barring Lebanon), they have enough of a relationship with Egypt to at least usually keep the Suez Canal open.
Morocco is a Spanish puppet whose Sultan spends as much time in Madrid as he does in Rabat, and he even has his own Moroccan Falange.
Algeria is an orthodox fascist state with an utter hatred of the French. Very enthusiastic German allies, and have nasty race laws against the Tuareg population.
Tunisia's "nationalist" government is a joke, with its Presidents willingly selling out their country to Italian interests for a hefty sum. Basically an appendage of the Italian empire.
Egypt's authoritarian but pro-UN monarchy was deposed just before the Suez Crisis. The local fascists are Egyptian nationalists, not pan-Arabists, which infuriates the Syrians.
Israel and Lebanon are very paranoid but free UN-aligned democracies. Considering the Egyptian government, they won't give up the Sinai in a million years.
The Kingdom of Arabia is an absolute monarchy, like OTL Saudi Arabia but less aggressively religious. They don't like the Germans, but they pay better.
Yemen and Oman are dictatorships without much in the way of ideology that have bounced around between the UN and Axis camps over the decades.
Syria...er, excuse me, the United Arab Republic, owns Jordan. The Arab pan-nationalists in the Arab Party are frustrated with the rest of the region's hesitance to get on board, but anyone seeing how the Syrians run Jordan wouldn't be very enthusiastic.
Iraq is run by the Ba'athist Party, also Arab pan-national fascists but with some quasi-socialist tinges; the Strasser to Syria's Hitler. Mainly, they just think they should lead the Arab project instead of the Syrians. They fight constantly, and Syria usually wins.
Iran: After the Shah was ousted in an OSS-sponsored coup when he tried to cozy up to the Germans, Iran was briefly a democracy before being taken over by a left-nationalist junta. Seen as an important UN ally in the region, Iran democratized in two bursts, first in 1980 and then in 1987 when the junta relinquished full power. Now on their Sixth Republic (the junta changed governments thoroughly), the Iranians are a rising economic star and a growing military power as well.
Sub-Saharan Africa:
South Africa is an apartheid state that was kicked out of the Commonwealth in the 1960s. It does own Botswana, and the occupation of it isn't pretty. South Africa isn't directly in the Axis, but they still keep the UN out of the region and that's good enough for the Germans. Angola, Rhodesia and Mozambique are all very, very nasty apartheid states that are about as progressive on race as Germany, if not worse, and South Africa uses them to bully countries it doesn't want to be caught interfering with itself. Zambia, under the rule of a nationalist but pragmatic President-for-life, plays along with the South African gang to avoid getting trodden on. Malawi is a very, very, very paranoid country, but they haven't been conquered yet, so they must be doing something right.
Tanzania broke back up after a few years, and Zanzibar is both pretty prosperous and one of the few genuine democracies in Africa. The Sultan has built casinos, clubs and all kinds of amusements to put Las Vegas to shame. Tanganikya is really regretting splitting back up these days.
The Federal Republic of Senegambia is a democratic republic, and about 20 years away from First World status.
I think enough has been said about Ethiopia elsewhere, quite frankly.
The Central African Empire is run by Bokassa I, who is getting on years, but has a lot more military success to back up his claim to being Napoleon's successor. He was armed by the British and French early on as one of the few African rulers willing to take an explicitly anti-Axis line in the mid-70s, but lately has tried to strengthen the Non-Aligned Movement. He managed to conquer Chad and later Cameroon as they fell into civil war, and is getting ready to pass the throne to his son. Central Africa is pretty well developed, considering, and is the biggest sub-Saharan African power after South Africa and Ethiopia.
Asia:
The People's Republic of Afghanistan never approached the totalitarianism of Stalin, but it has not liberalized to the extent of the modern USSR, either. With the Soviets and Manchurians both deviating from the hardline communist model in various ways, Marxism-Leninism-Tarakism is the new hipster vanguard of international socialism. In fairness, there's equality for women and the cuisine's pretty good.
The Mongolian People's Republic has always jumped when the USSR said "frog," and liberalized with it as well. Actually allows some opposition parties to run nowadays, but the Communists are still in control and their economy is actually more state-owned than the USSR itself nowadays.
Japan, like Germany, had a well-developed exotic weapons program. Mecha were instrumental in the Battle of Hawaii, in which the Japanese captured and held the Hawaiian archipelago for close to six months in early 1942. After a brutal island-hopping campaign, the Americans landed on Taiwan in late 1946 and Okinawa in early 1947. A void bomb dropped on Hiroshima, on the condition that Japan would keep Taiwan and the Emperor. The Americans agreed, seeking an end to the war themselves, and thoroughly de-militarized Japan and more thoroughly eliminated the war criminals in the Japanese government. Since the UN had no need for Japanese rearmament, the country remains without either an army or a Self-Defense Force, the only men under arms being the National Police Force. Along with Korea, however, Japan would reindustrialize with rapid speed with the help of American contractors during the Manchurian and Vietnam Wars. Today, Japan is very rich and very pacifist, with a great deal of transitions of power between the Liberal Party and Socialist Party. While the Chinese still claim Taiwan, most issues were solved in a 1993 agreement creating a devolved Taiwanese Parliament and allowing for free travel of Chinese nationals to and from Taiwan.
Korea is united, and after a period of rule by strongman Syngman Rhee democratized under its Second Republic in 1960. A middle power militarily and a first-rate one economically, it is very close to China politically and advocates a war against Manchuria to remove the regime's threat.
Sri Lanka is a Dominion, and is decently close to London still. Gandhi is a notable citizen here, exiled by Bose during his revolution, and his words were instrumental in preventing a civil war between Tamils and refugees from the Indian mainland. The British eventually made their peace with him, and he spoke out against Boseist India until the day he died.
Thailand is involved in the Non-Aligned Movement, and while it is a kinda-sorta democracy it is suffering a serious case of what we would call "Finlandization" with respect to India. Coups are frequent.
The Democratic Republic of Laos and Socialist Republic of Cambodia are impoverished Soviet satellites, sure, but at least they've avoided civil war and genocide.
Vietnam was granted independence by the French in 1948 as the Empire of Vietnam, which was couped by military officer Ngo Dinh Diem and transformed into the quasi-fascist Republic of Vietnam. Ho Chi Minh met with President Stephenson of the US, who orchestrated a joint Franco-American invasion to quickly topple Diem and install Ho Chi Minh as President. Vietnam has developed remarkably well, and is only slightly poorer than Japan, China and Korea. While the Communist Party (really more social-democratic these days) almost always wins the elections, they aren't jerks about it. Saigon (Uncle Ho had a bit less of an ego TTL) is a happenin' vacation spot.
The Dominion of Malaysia isn't that friendly with the UK these days, but they still have the Governor-General and aren't making plans to get rid of him any time soon.
Indonesia is the only Asian founding member of the Non-Aligned Movement. It has gone through a few different governments, but the current dictator has instituted a hybrid of nationalism and Randism that is actually industrializing the country with some success. Ignore those forced labor allegations, that's just imperialist-socialist-fascist propaganda.
The Dominion of Australia has had an alarming number of quasi-fascist governments in the post-war era. The National Party, a bellicosely nationalist and monarchist outfit, is currently the dominant party in the Australian parliament, and is doing everything it can to avoid letting go of New Guinea, which it invaded in the 1970s to oust a fascist-ish dictator and never quite left. The New Guinea Assembly is almost totally dominated by the Free New Guinea movement, but Sydney has been coming up with excuse after excuse to postpone the referendum. Between this and the passage of a sweeping new security act that greatly expands the government's surveilence powers, there is increasing pressure on the British to tell the Aussies to knock it off.