I have sent animals on BA and other airlines from the UK, but not cats from USA to UK. I get the impression that the ground handlers at LHR are extremely good, and many of them own dogs of their own, so are often checking after the animals out of both curiosity and a real sense of care. You best use a specialist importer in my view, who then liaise with IAG Cargo. You may also want to consider airlines like Air France, which allow you to take cats with you in the cabin, and then using the tunnel to the UK. Generally animals are best in the hold but equally they are reassured if they can see their owner steward.

There is a temporary restriction on carrying dogs (not cats) on BA which has been partly lifted (MAD, DUB and the UK), but currently for the rest of the world BA are not taking new bookings for dogs.