We sent our dog from EWR to LHR in 2012. I called all the airlines that offer pet transportation and spoke to them, I also called and spoke to the company at LHR that coordinates pets from airline to ARC. They were very helpful.
Ultimately we went with United (most costs were the same). Followed the rules laid out by US and UK, which was a crazy timeline of forms, injections, forms and stamps. We had to provide the crate, labels, bowls. You don't pay until you drop the dog off at the airline, which was strange. It cost us a lot of money and it made me very stressed. The dog was fine though. She arrived in Heathrow, was taken to ARC, given a meal (I had SPECIFICALLY asked/written that they should not do that, she's allergic to everything) and then was collected by my Father. Happy as a clam.
I can try and dig up more of the detail from when we sent her (I posted about it on another expat site) if that helps? I know a lot of my clients who move internationally use the companies who take the dog from your house and handle it all, but I'm a control freak, plus I wasn't moving, so seemed easier to save some $ and do it myself.
(For clarity, she is a 55lb Boxer, it was supposed to be temporary, but my parents refused to give her back, and she's living happily with them still, aged 12.)