We fly in three weeks for the first time in First class to London with our baby (he is 1 year old).

Few questions for the experts:
British Airways advertises on their website infant seats. Are they comfortable to sleep in for babies 1 year old? Are they completely horizontal like a bassinet, can a baby sleep for 6-7 hours straight in them? If we end up buying a seat for our baby (full price), is it possible/safe to have him sleep in the adult seat (we are worried he might fall off the seat while sleeping)? Any other tips for traveling with baby in First class?

Thank you