There’s a few threads asking why Bristish singers sound like American singers or sound like they don’t have British accents or whatever combination you want.
How about some British singers who sound British?
Al Stewart - “Time Passages” is a good example
Kate Bush (who, BTW, is extremely late in her follow-up to The Red Shoes. C’mon, Kate, we’re waiting.)
Can you think of some others?
Don’t know where Avril Lavine (sp?) is from, but she sounds different, as in not from Texas
Robert Smith of the Cure
oh, and Avril’s Canadian
The Beatles.
Lindisfarne.
Some of Rod Stewart’s older stuff has his accent showing thru.
Don’t know if he’s Aussie or Brit, tho
Almost all of the British punk rockers.
Rod Stewart’s definitely British.
Some others:
Johnny Rotten
Syd Barrett
John Cale (if you count Welsh)
Well, if you want to go way back…
Davy Jones of the Monkees and Peter Noone of Herman’s Hermits were both from Manchester, and it showed both when they talked AND when they sang.
I mentioned this in one of those other threads:
Anthony Newley.
The Smiths’ Morrissey (also from Manchester)
The Cockney Rejects
The Macc Lads
Ian Dury and the Blockheads
The Pogues
Joe Strummer from the Clash.
Madness
Billy Bragg
Liam Gallagher (Oasis)
Andy Partridge, Colin Moulding
Damon Albarn (Blur)
Jarvis Cocker (Pulp)
Sophie Ellis Bextor
The Proclaimers - emphatically Scottish.
Your man from the Specials (Neville something?)
Paul Weller (The Jam amongst others)
Ian Brown (The Stone Roses)
And a rake of others
And Rod Stewart’s a proud Scot. Whether Scotland’s proud of him is another matter.
Not sure the Pogues are in the U.K. 
David Bowie USED to - on his much earlier stuff, you could hear the “Saff London” accent.
Also - most, if not all, of U.K. folksingers - but I suppose this is about pop/rock stuff.
Robbie Williams.
The late, lamented, much missed Ian Dury.
AudreyK: Robbie Williams? Cite?
Tansu: Agreed. New Boots and Panties is a musical masterpiece.
(doubt if anyone will have heard of these, though)