I think that the UK, with all those resources (from the UK, from France etc) I think that Portugal (which was "fascist") and Spain also join the axis, as well as the whole Middle East ... the axis does not he would have difficulty beating the USSR having all the oil he wanted and the rubber. As for accepting the government of Mosley ... it depends, I want to remember that shortly after the fall of France, the population of London took to the streets to ask for peace. If the UK falls, there is a huge chance that the population will take to the streets in the colonies to make peace because the war is lost. Churchill struggled and kept the country on its feet, when it was alone against the rest of the axis and only the military victories in North Africa gave hope, let alone with the "home islands" in German hand, there you go down to the square with "the war" written on it is over! ", in southern Italy (there are testimonies from my friends' grandparents, because I lost them too young) and in Italy in general, the" Anglo-Americans "were welcomed, not because they were" liberators "but because it meant one thing: "the war was over". Similar happened also to the Germans when they attacked the USSR. The Poles under Russian territory, and the Ukrainians, welcomed the Germans as "liberators from the communist yoke", I remind you that if all hope is lost, realistically you don't send your best youth to die (unlike the AI that also joins nations that are close to capitulating ...)
It's important to distinguish between the "colonies" like Malaysia, Nigeria or India, and the Dominions (Canada, Newfoundland, New Zealand, Australia, South Africa). The former will have little say in whether the war continues or not (although their governments might decide to side with the puppet government, that's unlikely for reasons I've described earlier), as they are explicitly not self-governing. Now, the populace is not negligible (especially India, where political concerns will be a problem keeping support for the war effort, but that won't translate into support for the Axis), but it's also not the deciding factor (any more than it stopped the "Free French" from recruiting heavily in the French colonies, which provided the vast majority of their pre-1944 manpower). The Dominions are autonomous democracies (with varying degrees of representativeness) and will have to answer to their governments, but their governments (especially New Zealand and Australia, who would recognize a Japanese force in Southeast Asia as a direct threat) are going to be extremely unwilling to hand over former British colonies to the Axis. They also are not particularly affected by the war at the moment; Germany can't invade any of the Dominions, and while there have been some losses, they haven't been nearly as many as were to come (especially if the hypothetical invasion of the UK is pre-Greece). So while the fall of the Home Islands will be a major shock, there won't be a great deal of war weariness as yet (in comparison, Italy had been at war for 3 years by the time of the Allied invasion, had lost its African colonies and a significant portion of its army, and had been under Allied blockade since it entered).
The Germans invading the USSR is a different matter; they didn't welcome the Germans because they wanted the war to be over (by the time the Germans invaded, the war for them had barely begun), but because they hated the Soviets and hoped the Germans would be better (and when it turned out they weren't, you saw one of the largest and nastiest partisan campaigns in history).
It's certainly possible that the UK government (the legitimate government, not some Mosleyist puppet) decides to negotiate for peace after the fall of the Home Islands, in which case the rest of the British Empire probably agrees. Indeed, it's precisely the hope for this that will likely cause the Germans to be reluctant to actually go ahead and create a Mosleyist puppet government (as that would complicate negotiations). But that would require a peace that essentially leaves Britain intact and independent (without an occupation force or German access to British materials), not one that renders Britain a German puppet (or allows Japan to occupy Britain's Asian colonies). In which case Germany gets no additional resources, still has to guard against a British reentry into the war (which they almost certainly will do if the US joins, in which case the US can use them as a springboard into Europe as OTL) and facing a USSR that is still probably capable of surviving Barbarossa (they had essentially halted the initial German advance before Lend Lease really became significant), while Japan is still without oil, trapped in a quagmire in China, and on the road to Pearl Harbor.
If the government refuses to make peace (or the Germans insist on terms that the British can't accept), then Germany can continue to occupy the Home Islands, but the rest of the Empire remains hostile. And frankly, there's very little on the Home Islands proper that the Germans would want. Great Britain is not particularly blessed with natural resources (other than things like coal, which the Germans already have plenty of), and can't even feed itself (already a problem for the Germans). While the factories might be of use, the record of all sides in terms of using factories in occupied territories is not particularly good: the populace is hostile meaning you either have to import your own workforce or deal with uncooperative local workers (slow and inefficient at best, actively sabotaging at worst), the factories use different methods/equipment (and in the UK, a completely different system of measurement), and produce different types of equipment (which is a logistical complication for a German army that is already a logistical nightmare).