Do BA have special animal cargo handlers for pet birds transfer via Heathrow to Miami from South Africa?
There's more criteria there about what you need to do.
However, much of the information relates to cats and dogs.
It's worth noting that British Airways uses IAG Cargo for pet carrying. Their page on live animal cargo indicates that they'll carry almost anything, from snakes to birds to horses, and have information on pet container requirements. I suspect if you don't have an applicable container, they'll be able to rent you something - it'd make sense to have that as part of their service as it'd be a common requirement.
Note that there's usually customs regulations on transporting live animals, even birds, so it's worth contacting BA or IAG and making sure you have everything you might need - paperwork, vaccinations (do they vaccinate birds? I don't know) and the correct transportation container.