A few months ago I successfully got my small dog from NYC to London Heathrow.
The process is a pain for me but it worked out and now I am back to deal with my own visa.
I had to find a USDA accredited veterinarian, cost me loads, more then the original place I usually take her to. I learn that the vet might not know everything about the movement of pet to another country. So I had to rely on myself and the research I did. This site had helped me with questions big time! The important thing is to follow every step and make sure it's done correctly. The USDA Vet is located near JFK, no bus or train get there so can only get there by car. It's a small office on the ground floor. It's a quick process, the USDA Vet took a look at the paper work, paid for it, and stamp stamp and out the door. I used Virgin Atlantic to transport my dog to UK, took my dog to the airline cargo area on the day of the flight. My dog arrived safely.