if we take it that Grimm's law is right about gw <-> w <-> q
we get that
Celtic, Gaelic, Walloon, Wales, Cornwall, Gallic, all refer to the same tribe of people.
Obviously, the rooster association with France is because of the Latin stem gall- in gallus, meaning rooster, compare to the apparently unrelated pollo meaning chicken. It's possible that we have a p/q swap here, like penta/quint- (5), in which case gall-/poll- might be related.
At any rate, this leads me to suggest that the Celts from France to Ireland originally called themselves *Gwal-, and my guess would be that it meant 'people', since that's more or less what other tribes call themselves, e.g. Bantu/Botho (people), Thiud/Deutsch/Dutch (people)... etc.
Obviously, there's a different tribe of Gaelic speaking people that take the name 'Brit' - as in Britain, Briton, Bretaigne, Brittany.