No, they would have had US aircraft with US pilots in them.
BTW, did you know that there were US airborne tankers air refueling British aircraft during the war?
That was sort of the point for my asking the question about the ski-jump/ramp. US naval aviators don't have the benefit of it, but the Brits use it with their Sea Harriers. More than likely they would have had a VMFA detachment & just painted roundels on the jets. The pilot/aviators would be called "exchange pilots". No need for a shake & bake version of the AVG/Flying Tigers.