TIts possible that they carried their love of the Arthurian legend with them and bolstered the tale's popularity on the island.
I doubt Arthurian mythos played a role there : before Monmouth popularized it in the XII century, it remained essentially a welsh account whom first mentions appears (IIRC) around the IX century.
Bretons forces accompanied William in his conquest, essentially for the same reason flemish ones did : there were lands to take.
It's not that much different from Norman forces going in Sicily on this regard.
It doesn't mean that the Duchy of Brittany was an ally of William : at the contrary, if Conan wasn't poisoned, he wpuld likely have attacked Normandy in 1066.
So, what if England were conquered by the Bretons instead of the Normans? Obviously, there would need to be some prior PODs, as Normandy was much more unified and powerful than Brittany was.
You'll need more than political unification.
Brittany was relativly unified at this time (less than Normandy, but as well not having known as much trouble than it), but their rulers didn't have a claim on England, when William got one from the special relationship between Anglo-Saxon and Norman court in the early XI century.
Your first step is to get one, possibly by having the confessor living in exile in Brittany rather than Normandy. For that, you need to nerf Normandy.
It shouldn't be hard to find a PoD, but really more to enforce it : in 931, Breton chase the Norses off Nantes but Normans went to help them and defeated the Breton army.
If Guillaume Longue-Epée is defeated and killed in battle, french kings would probably not give
"maritime Brittany" to them, effectively restricting Normandy around Rouen and making Brittany keeping the other part.
If Brittany keep its shit together, maybe Mathuedoï and his son Alan II
, instead of dukes of Brittany, recognizing the french suzerainty, could hope for becoming princes or even kings of Britons with a more vague vassalage possibly restricted only to eastern and maritime Brittany.
- England/English becomes more Celtic (assuming those names are even retained).
I don't think there was a real chance of that : many bretons nobles were no more celtic than their norman or angevince counterparts at this point and would be even more if Brittany keeps its maximal extension.
Admittedly, you could end with a Franco-Breton elite that would introduce some elements, but as much than franco-normans introduced norse element on their own (aka, really few).
- England might be less embroiled in French politics, since the English kings would not be simultaneously dukes of French territories.
All depends on how much Brittany do well : OTL, Brittany *did* recognize french suzerainty, the link really going down during HYW at the same time than Burgundy.
Less territorialy good (even if it's an important point as I tried to point out above) than internally : one of the reasons Alan II recognized french rule was that he needed legitimacy to rule on Breton vassals.
Bretons rulers need to do good and in expansion and internally to get really rid of that, and that's not a given.
Furthermore, they would be embroiled in continental policies, would it be only because french great nobles and king would still try to grab parts of Brittany as they did practically since the IX century. Willing or not, they would have to deal with that.
- Brittany might remain independent of France.
I doubt it eventually, save a total desintegration of the french kingdom : men, ressources (Britanny being really less wealthy than OTL Normandy) were on french side eventually, and an alliance with nobles (particularly angevine, as their union with english rulers is likely butterflied) could take on Britanny relativly easily.
The internal divison of Britanny (one can unifiy the whole peninsula, uniting it is more hard) is going to be an issue.
To resume
1) Bretons defeat Guillaume Longue-Epee, that dies, and keep maritime Brittany
2) Breton rulers managed to keep the whole thing more or less unified, depsite recurrent troubles
3) A special relationship is established between Anglo-Saxon England (OTL, some Bretons exiled went there, so not that far fetched) leading to a claim being transmitted
4)When the opportunity arise, must have combination of these elements : peninsula relativly unified, no war ongoing with neighbours, enough ressources, and not defeated on battle.
is needed for the OP (for the outcome of the conquest, I tried to answer them), hope it fulfill what you asked for.