Maybe he can't actually speak English and his UT translates his French into an English accent, which confuses the replicator when he goes to order a suitable French beverage
But that's just it. Picard shares NOTHING with being French. Okay, sure not all Frenchmen drink wine and eat croissants while wearing white face-makeup trying to breakout of an invisible box. But one thing French people do not do is sound British! Which Picard very much did. There's any number of things we could speculate about Riker's accent not being Alaskan, Beverly's not being Irish, Geordi's not being African and Worf's not being Russian. We could probably surmise any number of things about that to give them their more neutral accents. But then that's still leave us here: Why does Picard, a Frenchman, sound British? Did all of them just happen to go to a school and serve on ships and spend their entire lives in places with neutral/mid-wester accents that washed them away and Picard just happened to spend his entire life in places to give him a British one? Or could this just simply be a mistake and oversight by the creators? I go with the latter. We can try and to rationalize it all we want in-universe but it's hard to not come up with one that doesn't really hold up to much scrutiny. (Especially if, IIRC, Picard's family had British accents as well.) It's a big error, plain as that.
The simplest explanation, given that Picard's family (even the little kid) and Louis all speak with British accents, is that British is the preferred accent in France in the 24th century. Either that's how it is taught, or - much to the dismay of modern French people - that's the language on the continent. A LOT changes in four centuries. Also, why is Beverly's or Geordi's accent more "neutral" than Picard's accent? What does neutral even mean in that context?
First off, you can never apply an absolute generalization to any entire population. There's always a bell curve. In high school, I knew a girl who was from somewhere in the northeast US, maybe Baltimore (or was it Delaware?), and she had a gorgeous English accent because she'd been raised by an English nanny. Second, I'm sure any French person who had learned to speak English well enough would speak it with a British accent (given that they probably would've learned British English, given the proximity). Sure, some people will speak a foreign language with their native accent, but those who have a good enough ear for accents, or have done it long enough, will be able to speak the foreign language without an accent. Third, it's four hundred years in the future. Even today, Europe is far more united than it was fifty or a hundred years ago. Britain and France still consider themselves very separate cultures, but I'm sure that a couple of centuries ago, New York State and Pennsylvania considered themselves separate cultures. Things change. Who's to say that 24th-century Europe isn't far more culturally and linguistically united than 21st-century Europe? Fourth, bottom line, it's a TV show. Picard has a British accent because Patrick Stewart has a fantastic British accent and nobody wanted to mess with perfection. He was portrayed as French anyway because it was a TV show and poetic license exists.
How many French people do you know? That's exactly it. As with most Europeans, I was taught British English by a teacher with a British accent. I still have a thick Italian accent, due to the fact that while I use English daily, I speak Italian much, much more than I speak English. But I am fairly confident that if I were to use English as my primary language, I will increasingly lose my foreign accent, and be left with, guess what? British English.
Picard's mother Yvette has a distinct French accent in WNOHGB and her maiden name Gessard, while obviously French is widespread in the midlands of Great Britain. On the other hand, Jean-Luc's father Maurice has a British accent "Tapestry" is British and the Picard. Too bad its not the other way around. It would easier to explain Jean-Luc's accent.