Being from Britain, and all though I don't view the current popular music charts very often, it's hard to find very British artists.
Bands I like that are British and amazing are:
The Arctic Monkeys
M.I.A - She is originally from Sri Lanka, but is based in London.
The Kooks
The Fratellis - They're new album is brilliant, just it's never played that often, they kinda dropped into the background after their general introduction to fame.
Paolo Nutini
The xx - Brand new indie band, Stars is a fantastic simple tune, great bass.
Hot Chip - You've most definitely heard Over and Over
The Rolling Stones
The Clash
The Specials
The Sex Pistols
Biffy Clyro - Check out their old stuff, it's far better than their latest album.
Pink Floyd
There's definitely more but I haven't the faintest idea of what they are right now. I prefer the old British music scene to the new one, although I'm 16, I can definitely tell it was much better, I don't need to do much comparing. A lot of what I listen to is more European, as I'm really into the techno/electro scene
But honestly, British music can be great, but it's really faded out a lot.