Creative Assembly most immediately spring to mind. You’d want at least one Total War game on there - Warhammer 1/2 is “double-British” but both are very recent - if older probably Shogun 1 or Medieval 1 or 2.
Lionhead would need to be on there with a Fable or Black & White.
You’d want one of the wide array of incredibly boring British-made driving quasi-sims people like for no apparent reason and can’t explain why they like (I’ve asked). Take your pick - whichever one has the most pop-culture recognition.
Rocksteady with the Arkham series is an obvious one.
Rare - Goldeneye or Perfect Dark or probably Donkey Kong Country.
Frontier Development maaaaaaybe - they’re relatively recent but I bet more people have actually heard of/played their games than a lot of British companies.
Whoever the hell it is makes Football Manager with, yeah, Football Manager, because let’s be real, that’s one of the few modern British games that goes beyond just “gamers”.
Makes me puke but the LittleBigPlanet guys maybe? Loathed EVERYTHING about that game but it was big, huge even.
Free Radical with Timesplitters?
The No Man’s Sky people? (puke!)
Ninja Theory are amazing but I dunno if any of their games are actually big enough to have any pop culture footprint.
Rebellion keep making games but they’re all kind of mediocre and forgettable (I say this as someone who owns countless Rebellion games).
There’s loads of late 90s through today stuff I’m forgetting I’m sure, too.