What happened to the British language?

You mean pre-Celtic, ask about local kentum dialects?

Besides that, I doubt that those languages vanished mysteriously since the history of the British Isles and its inhabitants isn't really in the dark.

Language change tends to follow occupation by dominant (invading) culture since it is socially advantageous for natives to conform. In addition, the dominant culture often suppresses local language because they associate its use with rebellion.
Much of this, possibly regarding the specific language you mention, can be found in the very readable ' Cambridge Encyclopedia of Language' compiled by David Crystal.

Where I live, the Bructeri were the local Germanic tribe. Then the Saxons came here, then the Franks ...
I don't know what dialect/language the Bructeri spoke or people living here before these arrived. But certainly every of those invaders changed the language.
But place names, for instances, didn't change a lot, not here nor on the British Isles, if they were in common use. Family names stayed as well (my family name is Old Saxon).

Like latin - It became redundant.