All four major powers did; but the numbers were limited for
What if the major combatants had decided to use Africans and Middle Easterners in much higher numbers then they did OTL in the course of the Desert War? ?
All four European powers did; but the numbers were limited for a variety of reasons, most of which came down to the European powers not wanting to do so for reasons of imperial politics.
Italy raised both regular and irregular units among the (ethnically) Moslem population in Libya and Italian East Africa, and some of the Christian population in Ethiopia, but remember, the Italians had fought two very expensive colonial wars with some of the "native" Libyan and Ethiopian populations; the British, in turn, recruited some of these same men (the Senussi, largely, in Libya, and did the same in Ethiopia/Italian East Africa) but because of Egypt's legal neutrality and a desire by the British and Egyptian leadership to not strengthen the army as a nationalist force, the Egyptian Army's duties during the war were largely limited to air defense and similar rear-area tasks. The British directly recruited men in Sudan and British Somaliland, but again, their service was largely limited to security and other rear area tasks, largely in the colonies where they were raised.
The French raised units from the North African (and to a degree, sub-saharan African and Levantine) populations for service, but in many ways, these were - essentially - "professional" soldiers recruited from what (accurately or not) were perceived as "warrior" elements of the population, not unlike the ethnically Indian elements of the British Indian army. The "locals" in French North Africa included a fairly small number of French citizens (essentially, Europeans or of European ancestry) and a larger number of Moslem who were not French citizens but were (essentially) "hired" as mercenaries, and the French were cautious about the mix; not unlike the British/Indian formations, there was always a strong percentage of "European" personnel and units in every such formation.
Germany scraped up whatever manpower they could, for obvious reasons, from disaffected POWs and other locals, but the numbers were always very limited.
In sub-saharan Africa, Britain, France, and Belgium all raised "colonial" forces, but other than those men who found themselves in the MTO/ETO with the French (and who were largely replaced by FFI personnel as quickly as possible) and the three British "African" divisions that ended up in SEAC, they were largely limited to "local" security and related duties.
In southern Africa, the British potentially had a large pool of manpower to recruit from, both in the British colonies and dependencies and (of course) in South Africa itself, but for obvious reasons, that did not occur. Getting the Union of South Africa into the war and keeping it there, even with the SADF's policies regarding non-whites, was more important to Britain's war effort than any potential "African" army organized along the lines of the Indian Army, which was considered - the political costs outweighed any military benefit, obviously.
The British territories in the Levant/Southwest Asia - the Mandate of Palestine, (Trans)Jordan, Iraq, etc. - all offered additional manpower pools, but the political issues inherent even in raising volunteer forces, much less conscription, were huge, and in Iraq, which was nominally independent, just like Iran, Saudi Arabia, etc., it just was not going to happen.
In terms of formations, the British Empire's order of battle included three divisions recruited by South Africa (which saw action beginning in 1941) and three more raised in British "African" colonies, plus the brigade raised in Palestine (which didn't see action until late in the war, 1945, IIRC); that was it.
The French raised one colonial division (9th) almost entirely from sub-saharan African enlisted men, and all or parts of about six other divisions, but a large percentage of the "non-white" personnel in all these formations were repaced by FFI or French citizens who could be conscripted under the authority of the FNCL government under the Liberated Manpower Program from 1944 onwards.
Best,