Why not have Brittany conquered by the French and have it's King re-establish his court in America? The idea of an exile government of celts who were from celts who exiled themselves from Britain amuses me greatly.
Well, that's not going to happen for a number of reasons.
Firstly, Brittany was a County, not a Kingdom. (Picky, I know.)
Secondly, the French take over of Brittany was fairly gradual, with the aristocracy coopted into the French system.
Still. I like the general idea.
PoD. Some Breton fisherman gets blown across the Atlantic, and a few years later manages to fix his boat and return home. This happens, oh, let's say about 1100 or so.
As the French encroach more and more, politically and culturally, more and more younger sons (who aren't going to inherit anyway) decide to flee to Tir newydd (or whatever the Breton version of that would be). A lot don't survive the trip and settlement, but over about a century, gradually a couple of thousand Bretons arrive in the New World. A few will have brought their wives, many will marry the local native girls, but a thriving settlement is established, and grows.
There's very little travel back to Europe, as it's pretty dangerous, but occasional journeys are made - people wanting to study for the priesthood, for instance. There's no economic resources in the New Land worth exporting back to Europe, particularly. Sure, a little dried/salted fish and some furs come back, but the fishing and trapping grounds in Europe aren't depleted yet, and the market's pretty poor so far. About enough to pay for some books, more animals, different seed, that kind of thing.
Relations with the local tribe start out good (otherwise the settlement would have failed), and the acquisition of iron working, draft animals and European grain means that that local tribe gains a significant advantage over its neighbours. Meanwhile, as the population grows, and grows it expands into what's now hostile neighbouring territory beyond the friendly group.
European rates of increase in North America tended to be about 3% a year, doubling every 25 years or so, or 16fold in a century.
By the time the 1500s roll around and ships get good enough for European nations to try to start projecting power across the Atlantic, the locals have been growing for some 300 years, and there's over a million local New Bretons, armed with iron and steel, and Christian. No way can any European nation conquer them.
With luck, and some good planning, they've continued to make alliances with (some) natives, and Breton becomes the lingual franca of the 'civilized nations'. Faced with e.g. France TRYING to invade, they formalize their alliance as "stadoù-unanet-XXX'. (XXX will NOT be 'amerika' as Amerigo Vespucci will never be born, let alone give his name to two continents.) Whether XXX is Breton for 'New Land' or 'New Britanny' or 'Turtle Island' or 'Joe's land' (Joe having been the guy to first made it back, and is listed in the history books as the discoverer) is up to the author of any hypothetical author of a TL fleshing this out.
How's that?