They may not have gotten as far as Britain, but one theory I’ve seen is that farmers from the Black Sea region migrated up the Danube valley after the Bosporus dam broke and flooded the region around the Sea. Same theory that others pushed east, ultimately to Tocharia. The time frame would seem to fit with this paper.
I like this idea too. (Also,spreads Indo-European language) However, Oppenheimer's DNA studies (mostly) doesn't support it.
The Tocharians were probably descendants of people who’d been there for a long while (see that recent topic about the Tarim basin mummies). Their language was fortuitously preserved in written form; it’s not too far gone to think that there were plenty of others in Central Asia which were not preserved, or some which may yet be found. Tocharian (A & B) is extinct, and seems to be alone on its branch — based on what has survived. Seems likely that there were a bunch of other twigs.
But yeah, the Danube farmers coming out of the flooding Black Sea basin (inundation took perhaps 50 years) is theorized by Ryan and Pitman.