Unfortunately you can't use UA to bring a pet into the UK. And unfortunately the 2 UA res clerks I spoke with didn't know that. But I called the UK gov DEFRA and was told that. Also, a pet into the UK has to be checked as manifested cargo - you can't just bring 'em along as luggage. Again, UA didn't know that. Their res clerks were ready to book me a flight and reserve space for my dog.

But since they were the only airline offering to check the pet as luggage for flights into the UK, I felt a little further investigation was necessary. The price is like $154 if checked luggage, but around $2k for cargo.