The Gaulish-Brythonic mystery connexion

As I understand it, there is a controversy over whether the Brythonic languages are derivative from continental Celtic or actually originated in Britain. This concerns me as my family has both a  mainland French and Breton background. The problem being, while I'm interested in learning Breton because of this, it's useless as far as solving this mystery is concerned. Since the Breton migration happened in early medieval times, Breton is likely the youngest form of Brythonic there is and thus the least helpful here. Truth be told, I don't know which langauge contains the oldest traces of Bryhtonic (if it's Cornish, good luck figuring this out as it's the most poorly preserved).

I myself do believe that Bryhtonic is linked to Gaulish, since the two cultural spectrums associated with these languages share a lot of Gods in common, in this case right down to the name, and something like that doesn't just happen. There has to be some historical explanation for it.

The other unknown in this equation is Welsh mythology. Welsh is considered a Bryhtonic language, yet Wales has its own set of legends that doesn't seem to reconnect with Cornish and Breton legend much (besides the character of Arthur, anyway, whom I don't believe is an Iron Age mythological figure in the first place), and a lot of people speak of "Welsh polytheism". So what's the deal here?

I'd be grateful to anyone who can find any clues as to the truth of this matter.