It seems the writers were a little confused at times.
Perhaps in his voice and some customs. There was an episode where Picard visited his family’s vinyard in France though. Picard also talked about being French a few times. One I remember is his fond description of the colors of the french flag, appropriately ordered blue, white, then red.
He was born in La Barre, France. English accent notwithstanding, he’s French.
By the time of ST:TNG, the *Second Hundred Years War *had been over for decades. There is no more “France.”
As is only fitting and proper on a civilized world. 
There’s also the episode where he says “Merde”, getting around the network censors and making all the watching nerds who took high school French squeal with glee.
Yea, I think the show is pretty consistent in claiming that he’s French. Of course his accent is British, but between then and now we have several hundred years, WWIII and the eugenics wars (or are they the same thing?), so it’s hardly inconceivable that parts of France have English speaking people who nevertheless consider themselves French
The Eugenics Wars took place in the 1990s and are seperate from Word War III, which was sometime in the 21st Century.
Why is it odd that a Frenchman can speak English? Or that he learned from teachers who had British accents?
Maybe the Universal Translator just made him sound English.
He just has his personal universal translator set on “Patrick Stewart” mode.
ETA: dammit
He was definitely meant to French. It’s mentioned a couple of times that he is La Barre, France (as well as shown in the episode “Families”.) He also speaks French in “11001100” (or similar). I recall reading an interview with Patrick Steward which said that the reason Picard doesn’t have a French accent is because Steward didn’t want to sound like Inspector Clouseau aboard the Enterprise.
I say that we revenge this slight upon the french by having someone with an outrageously silly french accent playing an English noble in the next ST series.
Although that sounds vaguely familiar. Something about ‘elderberries’, I think.
If you want to hear Stewart do a silly French accent, watch LA Story.
Don’t the TNG DVDs have test clips of Picard speaking with a French accent?
According to this Space Seed exchange they were the same.
Maybe time travelling Romulans made them split into separate events.
At some point, the official timeline separated them:
Of course, we also find out in “Family” that his brother, his sister-in-law, his nephew, and his childhood friend all have English accents too.
Picard was French. He grew up speaking French. He learned how to speak English from a really good tutor who both imbued him with both a love of Shakespeare and utterly eradicated his French accent, which is to say he sounds like an educated Londoner of his time.
Would it bother you if Stewart sounded like he went Harvard, Yale, or UCLA?
I once knew a lady who was French born and bred, and spoke English with an English accent. Really, anyone who learns a language at least tries (with varying degrees of success) to adopt the accent of a native speaker, and is it so implausible that the accent that Picard would emulate would be that of the geographically-closest English speakers?
I always thought he was a Canuck. I guess I confused his character with Kirk’s actor.