These are mostly places that rapidly decolonized in the 1940s and 50s in OTL, so I'm quite skeptical of Britain's ability to hold onto them in an Axis victory. Iraq, having already had a pro-Axis government for a short while during the war until the Allies retook control, is probably breaking away as in OTL. Same goes with Egypt; while it's still likely that the Allies would win out in the North African theatre, the fact remains that the British-backed monarchy was highly unpopular, so I don't see them lasting long. Free France, on a shaky foundation from the get-go, would have a much harder time holding onto what's left of its colonial empire than the Fourth Republic in OTL, so Syria won't be a French mandate for long either. Iran, given its rivalry with Iraq and that it was fairly democratic at the time, could remain aligned with the Allies so long as they don't undermine it or launch any coups like in OTL. The countries of the Arabian Peninsula would also likely remain on the Allied side, as would Kuwait. Algeria would likely be the political centre of Free France, with Morocco, Tunisia, and Libya in its sphere.
The Axis would of course be taking full advantage of the situation and would be courting Egypt, Syria, and Iraq, backing pan-Arabism and emphasizing the struggle against their supposed common enemies of Britain and France, communism, and zionism. On the other hand, while these countries' populations were largely anti-colonialist and anti-zionist, they didn't seem particularly pro-Axis, thus the pro-Axis post-colonial governments likely couldn't demand much of them in the way of sacrifices for the Axis powers (so Germany won't be getting much out of them for proxy wars with the Allies). Germany would need to use a longer leash than with Vichy France or Fascist Italy, and when it starts falling apart (as it's bound to eventually), their Middle Eastern allies would be some of the first to be looking for ways to cut them loose and/or stab them in the back.