Re turreted fighters, really they are an incredibly niche item, they can perform well in a few rather specific conditions but really are not worth the time, money and resources put into something that will basically only perform as intended on the Tuesday of every other month but only if the skies are clear and the sacred words have been said by all parties involved.
Putting a turret on a single engine fighter is an interesting idea, but really that's all it should remain, a thought experiment. Perhaps if the RAF was looking for something different then a larger twin engine attack aircraft could be developed and if anything that would be the one to be armed with cannons.
I must confess i'm probably wrong here but the F36/34 = Hurricane and F37/34 = Spitfire right? Perhaps instead of going for the hindsight influenced 20mm cannons, the RAF instead arms them with .50cal MGs, 2 - 3 in a wing. The UK had the capacity to make its own .50cal MGs with Vickers, so perhaps instead of going for the perfect option, go for whats available. Perhaps have a .303/.50cal mix, the initial Hurricane/Spit coming into service with 8 x .303s. But them having the mountings so that when the gun's developed and okay you replace 2 x .303's with a single .50cal and then the next ones out have quad .50's and no .303s. So by the time Mr Hitler and friends put on their romper boots, the RAF has a mix of the 303/50's and 50s only fighters in service (perhaps the initial one's with the 303's and 50's can't take quad 50's due to a different wing layout that would require too much work to fix or something).
And at the same time instead of a turreted fighter the RAF develops a destroyer of its own and F37/35 leads to the first cannon armed British fighter, but its a twin engine plane, if not the Whirlwind, something like it. Like the F.9/37 -
https://en.wikipedia.org/wiki/Gloster_F.9/37 And this in turn could lead to its more multi-role Beaufighter cousin (again swapping the .303's for .50s, perhaps 2 x .50cal 2 x 20mm instead of 4 x .303 and 2 x 20mm).
This would give the RAF two decent fighters that hit harder than before as well as one heavy 'bomber destroyer'/attacker aircraft which also gives the RAF its first viable experience with 20mm cannons and this can then shape future plans.
Also would this lead to the Fairey Battle either being canned entirely or significantly altered. And if so, might this not make the RAF look at the Blenheim, and then the specs for fighters coming soon, and have them make nervous noises and look at replacing the Blenheim with something else? Or at least improving it as best they can as technically its not in service yet (and hopefully at least not having the bomb bay doors shut with a bungiee cord, relying on the weight of the bombs to push them open!).