Start building fast destroyers and escort carriers. Convert some merchant ships if necessary. Try to get Canada to take up some of the load. Send some of those ships to do heavy patrolling in the North Sea and turn it into a graveyard for u-boats.
Problem here is the FAA is tiny, and even if you get it off the RAF (not a sure fire bet) You've got to expand the pilot pool, the facilties on land to train and support the crews and get them the aircraft. The RAF gave the FAA the bare minimum to survive and deploy and buggered up aircraft design.
To get more crews, you need the infrastructure in place to support them and allow the FAA/RNAS to grow again first. So once independence is won (lets say 25 by the time the treasury, naval and air ministries have stopped trying to strangle one another) what's needed is laying the fields for the future crops of pilots and aircraft to grow, churning out escort carriers etc in the early 20's to 30's will probably net you a number of ships that will be of little use by the late 30's because aircraft will have rapidly moved on and anything you've built in the 20's won't be big enough to take anything you're producing now as aircraft, possibly including Swordfish.
You'll need to get independence but not burn bridges either, maybe work out some kind of facility sharing for training and then have the FAA/RNAS guys go onto specialist training that the RN would provide facilities for. You'll also want to try and do joint stuff with aircraft and aircraft engines and let the FAA/RNAS work out its doctrine without the RAF going "You can't navigate over water unless you have two crew!"
And baring in mind this is well before radar is a thing, so the air wing of any carrier will largely be a mix of short range fighters, and longer range TBR type aircraft, at least until monoplanes start getting mature enough and aircraft manufacturers start coming out with suitable designs, all of which takes time.
I will say that I think that pre war, the FAA/RN's doctrine on aircraft use was correct. At the time, aircraft didn't really carry the weight of ordinance and there simply wasn't enough aircraft for the strikes for them to be decisive. Instead they would be used to scout, and slow the fleet, allowing for the surface ships to catch them.
The RN did this in the Med in WW2, and it worked out pretty well.
Otherwise fighters would protect the fleet and more importantly, deny enemy scouting aircraft the chance to spot the friendly fleet and get away, or to shoot down enemy aircraft that might be spotting for the enemy fleet and helping plot fall of shell etc. Which was another role the FAA had as they were also gunnery directors.
Whilst in the 30s the RN did practice putting Fisher's follies together as a full group, the lack of available carriers and putting all your eggs in one basket was generally seen as very risky. This could have given the FAA the numbers it needed if it was to conduct a massed strike, but here again the FAA instead operated even like this as waves so lots of smaller attacks but keeping the enemy spotted and keeping them under pressure was the goal.
