With no Battle of Britain, the British aren't going to suffer horrendous losses in both aircraft and pilots. So they will be able to retain a very large force of aircraft for home defense while still being able to reinforce imperial outposts. Mainly because production won't be dedicated almost solely to replacing losses in the BoB. I'm assuming production would be at roughly the same rate, making far more fighters available to station in places like Malta, Singapore, Malaya, Burma, etc.Why do you think the RAF would send home based fighters away?
I would expect much would depend on what kind of No BoB it would be but at the very least the Luftwaffe would have the same setup as they did post 1940 (e.g. some fighters for defense and fighter-bombers and bombers performing occasional nuisance raids).
Considering Britain kept a lot of troops and aircraft at home for defense and proper training while they were urgently needed abroad indicates IMO that home defense was their first priority. Even in 1941, when it became clear that the Luftwaffe had shot its bolt and was subsequently fully committed to Barbarossa, they still kept the bulk of the RAF in Britain.
It's easy to say the British would/should have released aircraft for other theatres but in reality the British wouldn't know what the Germans were up to and if they wouldn't change their minds and launch a major air offensive against Britain at a later date.
Getting aircraft/troops sent to reinforce secondary theatres back to Britain again would be time consuming so the Luftwaffe would have a serious headstart in such a scenario.
I think it is therefore very unlikely that the British would have transferred tanks, troops and aircraft in any significant number to secondary theatres as long as they perceived a genuine threat to Britain and it won't be until late 1942 when the Americans are ready to start propping up the tottering British Empire with American troops and ample lend-lease equipment that this outlook changes.