I think a "No infants in First (BA policy)", implying you can't bring an infant into First, is rubbish (let's have the near-inevitable arguments about whether small children should be in F or not somewhere else, we've all heard it before).
BA does say you can't use Amsafe seats in any premium cabin.
They also say
Carrycots
For your baby's comfort, we have specially designed carrycots on board, so that your baby can sleep during the flight. If you have reserved your seat at a carrycot position, you can ask the cabin crew for a carrycot.
The carrycots supplied by us are:
...
available on all long haul flights (except in the First cabin, and on the London City to JFK service)
which seems to imply that there are no bassinet seats in First, but that is not in agreement with much experience here on FT.