We brought 2 cats from the UK to the US, but it was years ago. I can tell you a few things - airlines don't provide crates, but they do specify size (based on the size of the animal) and it has to be an 'airline approved' crate. Those are available from lots of vendors, or you can make one to spec.
We called British Air, and asked the procedure. At that point in time, the cats were 'extra luggage' and the cost was minimal. There is no quarantine requirement for animals coming from the UK, but you may need to provide proof of rabies vaccination.
AFAIK, there is no need/requirement to use a pet service - to start, I would check the requirements for the state you are flying to. That will tell you what (if any) vaccines are needed. Call the airlines and verify their requirements. I am not aware of any airline flying INTO the US that allows a pet to be in the cabin. Amazingly, animals take this pretty much in stride. Water/food has to be available, and they do go in the hold. My elder cat was totally cool about the whole trip, and Mr. Nasty - my little Phil who was an abused rescue cat who didn't like anyone very much, was totally fine with the experience.
This may be a good place to start:
https://www.seatguru.com/airlines/British_Airways/pets.php