The British Airways pet policy is as follows:
Pets are not accepted in the aircraft cabin.
From the UK: BA has partnered with PetAir UK, which is run and owned by fully-qualified veterinary surgeons and has been flying pets all over the world since 2004, to offer BA customers a safe, comfortable and convenient way to export their pets from the UK.
To the UK: if you are looking to transport your pet to the UK BA’s sister company IAG Cargo can help. They have decades of experience transporting animals from every corner of the globe and will look after your pet as if it were their own.
If you’re entering the UK, your dog must meet the requirements of the Pet Travel Scheme offered by the UK Department for Environment, Food & Rural Affairs (DEFRA). It allows cats and dogs to travel between some countries and UK airports avoiding quarantine.
If you’re travelling to a different country, please check with the relevant embassy or consulate for any regulations or restrictions.
Taking a pet to a foreign country involves preparation and planning far in advance of your departure date. Requirements for pet travel to a foreign country are set by the destination country. If you are you traveling to another country and bringing along your pet you’ll need to meet the animal health requirements of the country you are visiting.
More information on the British Airways pet policy