It makes a lot of sense really. By Picard's time english is the standard language and probably spoken natively. Your accent comes from learning the language from those around you. The British are right next door, so they would pick up on their accent. Foreign accents come from having to translate your native language into the language you are speaking, the more comfortable you are, the less pronounced the accent will be. Case in point, many mexican or Italian americans speak english with their regional accent in English, yet also retain the accent in their ethnic language. A frenchman with an English accent 400 years from now is not so farfetched.
The question is not where you were born, but where you grew up.
Well, I think Gérard Depardieu would have been a good choice for Picard, but...
This is standard Star Trek practice really, isn't it? Apparently Uhura and Geordi were supposed to be from Africa, but they still had American accents.
The question is not where you were born, but where you grew up.