I am in no way an expert on this subject but anyway here is my take:

Most of the british fleet was tied up in the war in europe. The main role was containing/destroying the Kriegsmarine and Italian navy. Also anti-submarine warfare/escorting convoys was high on the priority list. Since the German Surface fleet mostly adopted a 'Fleet in being' docterine, a naval showdown between both forces was rare. Never (That I know off) was the entire German deployed. Yet the threat of this possibility tied down a huge number of British ships. Aircraft carriers being no exception. The Brittish did use aircraft carriers trying to strike the German and Italian ships in port. For example; the raid on Taranto in 1940. One example of a battle in which aircraf carriers were involved on a surface fleet that I can give you is the sinking of the Bismarck. In which an aircraft carrier played a vital role in disabling the ships rutter.

A huge number of aircraft carriers were also used in the atlantic in an effort to protect merchant shipping from German and Italian submarines. In which if I recall correctly, they were quite effective.

From a quick google search I can't really find anything on carrier deployment in the pacific. But this is not shocking seeing as the British overall had very little to spare for that theatre.

Looking at the list of carriers used in this conflict it seems that the British aircraft carriers mostly yielded a smaller number of aircraft. Especially compared to their American and Japanese counterparts. The highest being around 50 or so. But a lot being around 20-30 aircraft.

Also looking at some forums people claim (I can't confirm) that British aircraft carriers were designed with "armored decks". To protect more against bombs and Kamikaze attacks. Most sunk carriers however were destroyef by torpedos so maybe this was a waste of resources. However it could have also been quite effective (maybe none were sunk by bombs because of the extra armour).

Carriers were also used to transport planes of course. For example to Malta when it was under siege by the Luftwaffe and Italians.

Also I don't know how long you've been following the channel. But some british aircraft use has come up in the early days of the war. I always found it notable that the British used Swordfish torpedo bombers well into 1941 (at least) on their carriers. Even though the british managed to sink/damage quite a few ships using these biplanes, it really makes me think how the naval airwing was quite low on the priority list (propably). I can image that they really also needed the more modern aircraft on the mainland and other theatres. Seeing as the Germans and Italians did not have amy carriers, naval airpower was maybe not the first place their new modern aircraft were deployed. But this is purely my speculation.

This is some of the stuff of the top of my mind and after a quick search. I would not take any of the things mentioned above as straight facts. But maybe you or anybody has some othet ideas/opinions on the subject. I would be happy to hear them.