Yeah, there's that. Different broadcasters use different title music. It's not FOM's responsibility.
FOM produce the race itself and, before that, just shots of cars on installation laps/crowd shots. They produce the unilateral interviews as well (the guy that interviews the drivers is Peter Windsor, who is normally with the US broadcaster, SPEED). Not sure about after that. They also do the same deal for qualifing and free practice.
IIRC some countries (I think Canada is one) don't actually send anyone out, the commentary is done in a studio somewhere. ITV use the FOM feed in part (you can tell when it is as there is the F1 at the bottom) and their own stuff. Their own stuff is generally just Lewismainia apart from Martin Brundle's gridwalk.