AFAIK conventional wisdom doesn't tell that narrative about the Celts. The Celts are most often supposed to have arisen as a distinct ethnicity in Central or Western Europe, more probably with the Urnfield culture. The "core" Celtic peoples, those most often associated with typical Celtic culture, are often supposed to represent specifically the Hallstatt culture around the 9th century B.C., from which centuries later the expansionist culture of Gauls would sweep through Celtic and non-Celtic territories spreading the La Tène cultural styles (and probably Gaulish dialects themselves, which in my opinion is why the Celtic languages were suspiciously too similar for an IE branch supposed to be old).
If I were to make an "informed guess", I'd bet on this speculation: Indo-Europeanized Bell Beaker spread an ancient Northwestern Indo-European language north, south and west from its supposed starting point of expansion around Netherlands/Belgium. Spreading their mixed culture and language to the south through the Rhine, the Proto-Celtic culture would've arisen as a southern extension of that Northwestern IE next to the sources of the Rhine, in present-day Switzerland/Southern Germany/Alsace-Lorraine. That's also broadly where Hallstatt culture appeared.
I believe Britain was inhabited since the late 3nd millenium B.C. by Para-Celtic peoples, i.e. peoples whose language was related to, but not derived from Proto-Celtic proper (perhaps another unknown branch of Proto-Italo-Celtic). Then those peoples adopted Celtic languages when the "core" Hallstatt Celts spread their languages throughout Europe. It would've been a relatively uncomplicated feat for them, much like switching from Swedish to English.