It appears not, according to the British Airways website:
Making your booking and taking your flight
The DEFRA Export Section (Dogs and Cats) will advise you of the
current documentation and vaccination requirements for your
destination. We will also require a health certificate from your vet
to show that your pet is fit for air travel.
When you make your booking you will need to let your travel agent or
British Airways booking representative know that you will be
travelling with a pet.
Your pet will not be able to check in with you but will be handled
professionally by British Airways World Cargo, a cargo booking will be
required which can be requested through the local British Airways
World Cargo booking office.
On the day of travel, deliver your pet to the British Airways World
Cargo Office. The travel container will be checked and the appropriate
charges collected. Please check local payment options when making your