You might find useful some of the information in the British section of my G40 map analysis:
Egypt, for example, was technically an independent country during WWII, though there was a lot of British influence. It was also somewhat unstable; some members of King Farouk’s inner circle were a little too pro-Nazi for London’s taste, and in 1942 the British Army actually staged a mini-coup to get rid of them.
I agree that if the UK itself had been occupied the rest of the British Commonwealth’s self-governing Dominions, who in 1939 had declared war in their own names (rather than being automatically dragged in by Britian, as was the case in 1914), would have continued the fight on their own. The British colonial territories are a more complicated case, with no clear answer. India is particularly problematic to speculate about; if Britain’s homme territory had come under Nazi occupation, the independence movement in India would probably have been given quite a boost. Even if India had, however, declared itself independent from the UK, this would hardly have translated into India surrendering to Japan, since that would have undercut the whole point of being independent in the first place.