In the period July-September 1940 the British had five "combat" aircraft carriers available, as best I can tell. These were Ark Royal (54 aircraft), Eagle (21), Hermes (26), Furious (36) and Illustrious (36). In total that's 173 aircraft of all types. That's not really a big heap of aircraft. It's better than none I suppose. And I have to wonder whether the carriers would be committed or just the aircraft. There were also a few "escort-quality" carriers, e.g. Argus & Unicorn. It's very likely that the aircraft would have been taken off these ships.
The other problem with using the carriers is that the British carrier aircraft are not splendid, unlike, say, the Spitfire (the Seafire and Sea Hurricanes were developed later). You don't really think of the Blackburn Skua or the Fairey Fulmar, etc., as among the great planes to come out of the war, never mind the Swordfish. They did get the job done if the circumstances were right, but they were not brilliant planes. Even if the British managed to assemble all these carriers they probably would be good for only a few days of heavy combat. If you were going to commit the Fleet Air Arm, you might consider taking just the pilots and putting them in Spitfires and/or Hurricanes if possible.
The German/Italian failure to secure the Mediterranean really can't be put down to the British carriers, I don't think.
JR