British Airways does not carry pets in the cabin (assistance dogs aside); animals travel as manifest cargo through IAG Cargo, which manages crating, welfare and customs.
In-cabin travel
Not offered Checked (hold) travel
Not offered Manifest cargo
Allowed IATA Live Animals Regulations: IATA LAR (IAG Cargo)
Comparing carriers for a route?
Airline pet rules stack on top of the destination rules. Check both before you book.