You can fly into London with pets but they need to be in the luggage hold, whether airlines will allow this or not is up to them. Your pet will need a passport, micro chipping and proof they do not have rabies (potentially other things, you will have to research this).
Additionally, be aware finding a rental property with pets in London is VERY difficult. Most landlords do not allow even goldfish. They won't even consider your application if you say you have a pet.