I've been a big fan of medieval whodunits since the day I fell in love with Ellis Peters' Brother Cadfael series, so I guess this one can still be counted as my favourite (he's more Welsh than British, actually, but most of the stories happen in Shropshire, during the fight for kingship between Stephen and Maud. So I guess that makes it British enough). It's also been turned into a TV show of varying interest, depending on who gets to play Cadfael's sidekick and friend, Hugh Beringar (3 different actors for only 13 episodes).
As for "real" British medieval detectives, you can try Paul C. Doherty's three main series : those are :
- the Hugh Corbett series he wrote under his real name, and where the main character is a clerk to Edward I,
- the Kathryn Swinbrooke series, by C.L. Grace, my favourite, where the main character is a lady doctor during the War of the Roses,
- the Brother Athelstan series, by Paul Harding, where the main character is the assistant of the London coroner, at the beginning of Richard II's reign.
And of course there's also Kate Sedley's Roger Chapman, who's a... chapman, during the War of the Roses.
Oh, and she's not British, but I'm currently reading the Sister Fidelma series by Peter Tremayne, and she's a damn good detective too ! And the whole series is a wonderful trip inside 7th-century Ireland, and its very particular laws and habits.