I'm currently reading 'Partners in Command' about the interaction of Marshall and Eisenhower. The US was very concerned that Britain would not hold - US generals predicted defeat and British military leaders were very concerned that if Dunkirk could be done then Germany could pull a Dunkirk in reverse. The difference, of course, was that Germany did not have a navy in the Channel and Britain did.
Personally I doubt the premise - the Luftwaffe would have had to have far better intelligence and planning, and been far luckier, to shut down both the RAF and the RN. But that's the premise: Britain falls.
Here's what I think likely: most of all of the royal family and government relocate to Canada. A puppet government is set up in Britain and it demands control of the Royal Navy and the colonial possessions. They get none of it but Ireland is heavily pressured to join the Axis (no idea if that succeeds or if, like Spain, Ireland cavils).
The US sends military equipment to the Philippines instead of Britain and takes a slightly less hostile tone with Japan (probably not including an oil embargo since that was in error anyway). If Japan strike it might target British (Burma) and Dutch (Java/Sumatra) oil production and not the US, or it might go all-in as it di historically. If so it gets beaten worse and quicker - there is no 'Europe First' strategy and no pressing reason for Germany to declare war on the US.
Germany turns on the Soviet Union and, with Europe united behind it and any British oil supplies added in, might have a good shot at winning. Even if the US gets into a war with Germany there is no way to fight convoys from New York to Murmansk, so subtract all of that US fuel, trucks, radios and the like. At worst, the Soviets cut a 1917-style deal, give up everything through the Ukraine and promise oil in tribute. If the Soviets go down, the US cannot win a war against Germany. North African operations would not be enough; with the British Isles under occupation, Egypt falls and a revolution might succeed in Iraq.
If Germany gets a solid oil supply (from a beaten Soviet Union or Iraq) then WW2 is over and Germany wins a unified Europe. With years of quiet to prepare and no Eastern Front, no American invasion of Britain or North Africa will succeed. If Japan pushes the US too hard then Japan goes down. The fate of China rests on whether or not someone can clean out the corrupt mess of the Nationals (unlikely).
So... by 1944, Germany rules Europe except for tributary states like Spain and Sweden. Japan is either cautious with the US or dead, China is communist, the Soviet Union is a rump (possibly fighting the Japanese over Vladivostok) and the North/South America block stands alone.
The more likely point of departure, I think, is for Halifax to come to power instead of Churchill and for Britain to cut a deal with Hitler after the fall of France. Crushing the RAF would be incredibly hard for the Luftwaffe to do, and if they could it would require a long campaign with better planning, intelligence and command. Also smashing the RN and forcing it out of the Channel while an invasion was on-going... I strongly doubt it; the Navy frankly had better leadership than the Army and would just keep coming no matter the losses. Doesn't take much naval force to inflict disasters on an amphibious invasion.