Let's start, you've pretty much regurgitated the F.5/34 requirement word for word:
Bristol 133, built instead of the Gladiator, got quite decent performance for that era with a 640HP Mercury radial, given a more powerful version of the same engine, well, it'd be falling behind the curve by 1940 but would still be useful in second line roles.
Continuing with Bristol, the
146. 8 gun monoplane, radial engined (840HP Mercury)... top speed is somewhat inferior to the Hurricane I (290MPH at an unspecified height vs 320MPH at 17,500 feet), but given a more powerful engine (
Perseus as planned, or perhaps a late model Bristol Pegasus instead?), it'd likely be able to make up that difference.
The Gloster F.5/34 has already been covered by others. It's the best looking of the options, but top speed (275MPH at 16,000 feet) of the prototype leaves much to be desired. Again, possibly fixable given a move from an 840HP Mercury to late model Pegasus, Perseus or a Taurus, or serious redesign around a Hercules (not simple... we're talking about moving from an engine weighing around 450kg to one of 870kg).
Vickers Venom. Personally, I think it looks hideous. Still, has a top speed in the right ball park (312MPH), was supposedly fairly manouverable and carries the required 8 .303 guns... All on only 625HP. However, it used a light weight and compact
Aquila radial engine which was already approaching the end of it's design spiral... and all alternative British radial engines were rather heavier (requiring serious redesign work on the airframe...).
Edit: If willing to bring in foreign designs using British engines, the
Fokker D.XXI (equiped with a Bristol Mercury) or
Koohoven F.K.68 (in production form equiped with a French engine, but a Taurus power variant was planned) would seem like fair bets.