I recently read a book called "The Celtic Empire" -- I forget the author's name, and his reasons for the dumb title (they weren't convincing), but nevertheless it was interesting enough, tracing the places where Celtic people settled throughout Europe and parts of Asia, and even Egypt! (Celtic mercenaries who fought for the pharaohs were given districts to settle, and there are traces of Celticity in their descendants even today.)
Ireland and the Scottish Highlands were the only Celtic regions not conquered by the Romans, so we tend to think of them as the only "pure" Celtic societies in the world today. The Romans held Lowland Scotland for a while but didn't succeed in suppressing the Celtic culture there. Later invaders like the Scandinavians and Anglo-Normans didn't impose their culture on the society as a whole. At least that's the theory.
It could be argued that the Britons of England and Wales kept their culture even under Roman rule -- that Romanization only affected the top layers of society, and melted away when the Roman rule collapsed. Nobody is arguing that the Welsh aren't Celtic! This would also mean that the Britons who fled to Europe and established Brittany were introducing a relatively unadulterated Celtic culture to that region, which was already inhabited by more-or-less-Romanized Celts.
The Saxon and Danish invasions of England seem to have done what the Romans didn't in suppressing the Celtic culture in the Britons of England. The Celtic cultures in the rest of Europe knuckled under to Romanization and later invaders and lost all but traces of their Celtishness. This is likely a vast oversimplification -- but it explains why we automatically think of Ireland, Scotland, Wales and Brittany as "Celtic" regions even though at one time the Celts conquered most of Europe.
Sorry about the lecture, but the book is worth a look.