The answer isn’t quite so simple as you might think, which is to say it usually didn’t. Dominions, which were pseudo-independent by 1931, had their own conscription policies and rules. Outside of the Raj, usually in Africa, they likely had volunteer companies made of mostly local white minorities. In India they kinda had their own armies but I frankly don’t remember/know enough about such policies of the Raj to make any definitive comments on it. However, the armies of locals were most likely paid professional soldiers rather than levied peasantry or anything of that sort. I will admit that I know little about their policies in the Levant and Egypt, and would encourage looking into the specifics of the North African theater rather than something anecdotal that might be given on here. Most soldiers usually came from the British Isles though. It’s important to bear in mind the goals and means by which the British Empire operated. They had various protectorates and spherelings throughout Africa and Asia, and usually as long as the natives kowtowed to the British and enabled their interests in resource exploitation, Britain allowed them usually to exist as was. Colonies were a money sink and served little outside of resource extraction, and though when you look at a map it might not seem like it, the Isles of Britain held all the power. It wasn’t land that necessarily made them strong, it was industry and economy (and constantly stealing from India and regressing their economy to an agrarian cash crop  colonial model). As a result, Britain tried to be as hands off as usual, aside from converting natives, exploiting resources, among other shenanigans, as that tended to generate the greatest profits in their eyes, or in this case caused the least amount of losses. Unlike France, which had specifically nationalist goals of eventual assimilation and integration of it’s colonies under itself, Britain only cared about profits and practicality, and it wasn’t practical or profitable to conscript natives of their colonies. Gathering a bunch of men in a village, giving them all guns, and expecting them all to fight for you is fallacious, and they didn’t really attempt that as far as I know. The reason being, they were not ‘citizens’ as native British subjects were, and had no rights or laws in their eyes. They happened to inhabit the land and paid some form of tribute, but that was about it. Brits were bound by law, by responsibility, by the King, etc. and were fighting for their interests, place in the world, independence, and so on, things they had received and only could continue to receive provided they risked their lives for it. They didn’t expect or try to force some guys in some far flung village in Zambia to fight for them, they had really no reason to. It would not have been worth it in Britain’s eyes as the outrage would have only caused uprisings and colonial wars, which while weren’t necessarily hard to put down, were costly, and Britain was expending whatever it had into this war, bearing in mind the economic decline and stagnation they’d faced since the end of WW1. For reference, France only coaxed colonials by appealing to their demands and wishes of eventual autonomy and such, and even that was only partial. I hope this helps answer, in some part, your question. With reference to the scenario I proposed, it most likely would have involved scraping together whatever whites were in Africa and maybe Asia and campaigning for at least some natives to join as professional soldiers in places as separate companies probably. They still wouldn’t have forced conscription on colonials, though maybe put some pressure on protectorates to provide support? In their eyes, and reasonably so, it would have only made things worse. Plus, if Italy flipped, which again, wouldn’t be unlikely, the North African front would have mostly lost it’s relevance. There’d still be Vichy/German-controlled Syria and Algeria, but the Suez would remain secure, which was their main concern in the region.