The ansewer is probably much more complicated than you think.
The countries of the Commonwealth all responded to the war effort in different ways. The larger nations such as India, Canada, Australia, South Africa and New Zealand all sent their own indigenous armies to fight where ever they were required, so for instance, the enormous contribution of the Canadians in the D Day landings are a prime example. Most of India's contribution was focused in the far East, but they were also an important component of the Eighth Army in North Africa, and separately in Palestine. They had been trained in India and few if any came to the UK, and if they did they came as part of their own units for specialist training. An example of this is Indian Armoured troops some of whom may well have come to the UK while they were being trained on how to repair the newer tanks as they came into service.
Within Africa, the British colonies which had their own defence forces at the outbreak of the war were small in size and largely white. They became responsible for their own local defence, and as a result recruitment was stepped up, but I have no idea what proportion of the new recruits would have been black. They were trained and employed locally. Much the same picture applies in the Caribbean.
Undoubtedly there would have been individuals who made their way to Britain, or were perhaps already studying here, to join up. This was especially true of the RAF, so for instance someone from, say, Jamaica, with some prewar flying experience would have been welcomed by the RAF. In such cases the individuals would have been integrated into the British units they joined and trained alongside their white counterparts. There were no segregation or all black units like the American forces.
I think the best way to arrive at the detail you require is to look at each Commonwealth nation you are interested in, and trace their contribution to the war effort, from a domestic point of view. I would suggest you start with India, since that is the best documented, because the size of the Indian Army alone reached 2,250,000 men by around 1944, not to mention having a sizeable Navy. The Canadian, Australian and New Zealand forces were very largely white. As far as I am aware the South African forces were also mainly white, but the forces which were retained for home defence of South Africa and its neighbours may have had a proportion of black troops.
