The Celts moved into the British Isles in separate waves, one being the Goidelic branch (most notably preserved in Irish) and the other the Brythonic (most famously Welsh), and together they form the Insular Celtic branch of the Celtic tongues. But these migrations were relatively recent in the pre-history of the British Isles, and the Celts did not enter an empty set of islands but one filled with people speaking their own language(s). Even the more conservative Celticists admit there is a layer of a non-Indo-European language in the Insular Celtic languages that is demonstrable. It's demonstrable because sound correspondences allow the linguist to reconstruct its "proto" state, yet the word must be excluded from a native Celtic heritage for historical or morpho-phonological reasons. Here is McCone's shortlist:
*bratt- "cloak." Found in Old Irish brat and Middle Welsh breth-yn.
*mukk- "pig." Found in Old Irish mucc and Middle Welsh moch-en.
*we/aN-āl- "swallow." Found in Old Irish fannall and Middle Welsh gwennawl.
As McCone puts it, these words have a distinctly non-Indo-European "look" to them. Perhaps these words entered into the languages at an earlier time when the Celts were still together in Central Europe, when the Proto-Celtic language was still spoken, but the phonology of the three lemmatta above excludes a native root. And because there are no cognates outside the Insular Celtic languages, linguists like McCone assume the language of origin was the original language of the British Isles.
McCone, Kim. A First Old Irish Grammar and Reader. Maynooth Medieval Irish Texts III. 2005.