There was a great deal of anti-British sentiment throughout the Middle East. Middle Easterners were tired of being colonies of Britain and France. Initially, that sentiment would have allowed Hitler to recruit men for war against Britain. Later, those same recruits (and others) could be informed about the Soviet Union’s persecution of all religions (including Islam), and its repression of Muslims in the southern Soviet Union. A large force of Muslim men could invade the Soviet Union from the south, creating an additional front for it to have to deal with; not to mention entire armies that it simply didn’t have to face in WWII. Germany could supply this force with some jets and other modern weapons to improve its morale and military effectiveness.
If Germany had occupied the Middle East, I’m not sure Germany would have been able to recruit the local population to its cause (assuming Germany was inclined to try doing so in the first place). During the early days of Operation Barbarossa, some of the population groups in the western Soviet Union briefly entertained the hope that the Germans might prove to be more agreeable rulers than Joseph Stalin. The SS and the Gestapo soon came along and dispelled that particular notion. Germany was able to raise a certain number of troops in the various countries it occupied, but even with the help of collaborationist governments like those of Vichy France the forces asssembled in this manner were relatively small.
Similarly, the Japanese were never able to capitalize very much on the anti-British (and anti-French and anti-Dutch) sentiments that existed in the Far East. When Japan marched into one country after another in 1941-1942, it tried to market its conquests as a campaign for the liberation of Asia from white European colonial oppression. The conquered locals soon realized that they’d simply traded one kind of foreign imperialism for another, and that life under Japanese occupation was no picnic. Even Thailand, which was nominally an ally of Japan, was squeezed in a way which convinced pretty much everyone except the country’s top leadership that the proper response was to resist rather than collaborate. Japan also made efforts to cultivate the Indian nationalist movement, but apart from getting some support here and there from people like Subhas Chandra Bose it never got anywhere near to provoking a serious uprising against British rule.