He is Britsh automatically. Go to the British Embassy site to register the birth. Not compulsory but worth doing.
I´m British, my PACS partner Rossi is Bulgarian, we have our first child in France
He can get French citizenship at 13 or 16 providing he lives here for at least 5 years continuously (before 13) or 8 years (between 13 & 16). Nationality law varies a lot from state to state - many states clearly aren't at all sure just happening to be born somewhere to 2 foreign parents with valid docs (ie not displaced people or refugees), should confer nationality.