See if you can get this book, the answers maybe in it. https://www.amazon.com/Desert-Air-Force-World-War/dp/1844158179/ref=asap_bc?ie=UTF8
Until late 1942. There weren't that many Spitfires in North Africa until late 1942. By July 1943 there were more spitfires. Here is a link to the orbat of the North WEest Tactical air forces for Op Husky July 1943. My quick count is 21 spitfire and 18 P40 squadrons https://en.wikipedia.org/wiki/Northwest_African_Tactical_Air_Force
In Husky many American units were equipped with Spitfires. The P-47s or P-38s must not have been up to par yet for that operation. Too bad they didn't bring a few escort carriers with F4Us, I don't think F6Fs were ready until the fall of 1943.
The CVEs did not operate F4U Corsairs until very late in the war. For the most part the CVEs had the F4F/FM Wildcat, although later some CVEs did operate F6Fs..
Could they have landed P-47s and P-38s in the parts of Africa held by the British or were they not available yet in numbers or operation?
The USAAC did have some P38s in North Africa after Op Torch, but the P47 was a little too late to be deployed in North Africa. Deploying aircraft to "the parts of Africa held by the British" meant transferring them by ship around the cape or flying them across Africa by the Takoradi route. This took a while and required maintenance facilities and spares in Egypt, the base for British operations in North Africa.