Quite possible, yes, but since English is clearly the primary Federation language, it just seems probable to me that everyone on Earth would learn English as a first or second language from childhood. And it stands to reason that Europeans would learn British English, because it's closer to home.
It just occurred to me, though, that it's strange people only ask this question about Picard. I mean, there are other Trek regulars who come from non-Anglophone countries yet speak English with a perfect American accent -- Hoshi Sato is from Japan, Uhura is from East Africa, Geordi La Forge is apparently a Mogadishu native. Not to mention all the characters who speak flawless American English despite being born on other planets, like T'Pol, Spock, Worf, and Dax.
For that matter, why does Spock have a standard American accent with a Bostonian tinge when his parents have mid-Atlantic accents, and why do Worf and Nikolai Rozhenko have American accents when their parents have Russian accents? Okay, the latter can be accounted for by schooling; there are lots of cases where the kids of immigrants don't have an accent even though their parents do. But it's at least as big an incongruity as the Picard thing, yet it doesn't bother American audiences because we unconsciously assume American accents are the default. We only question it in Picard's case because an English accent sounds exceptional to us and thus demands an explanation. But since Picard is from Europe, it should be the other way around -- it would only be weird if he had a North American accent.