The British language actually survives as three separate languages today: Welsh, Cornish, and Breton.
Welsh remained the predominant language in Wales because it wasn't incorporated into an English-speaking state until more than 1,000 years after the Anglo-Saxon arrival in Britain. Cornish held out in Cornwall for much the same reason, although it eventually went extinct in the 20th century (it's currently undergoing a Hebrew-style revival). Finally, a group of British-speaking refugees from the Anglo-Saxon incursions migrated across the channel to the Gaulish region of Armorica, where they became the Bretons and the region became Brittany.
There was a fourth British-derived language in what is now northwestern England, Cumbric. It appears to have gone extinct by about the 12th century, but Cumbric-derived counting systems and other influences survive in the English dialects of the region.