I´m British, my PACS partner Rossi is Bulgarian, we have our first child in France

He is Britsh automatically. Go to the British Embassy site to register the birth. Not compulsory but worth doing.

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.