(I private messaged this info to the original poster but am posting here as well. For posterity, so to speak.)
AMERICAN vetrinarians are not totally up on the details of the UK laws and regulations. British vets are, oddly enough. Find one in the town you are moving to that will communicate with your vet. There are also animal transport companies that specialize in the process, we happened to find one of these.
I would recommend a company called Par Air Services:
http://www.parair.co.uk/ They have been moving animals in and out of the UK for decades. I found them by searching for catteries in Essex, and our cats finished quarantine at their facility. They will assist you to get what needs done at your end, and then pick up the pet and transport to you or a quarantine facility anywhere in the UK. Pets can't enter at every airport, and they are reasonably close to Heathrow and Gatwick, so your pet can be in quarantine there, or nearer to you.
(If recommending a particular company is not OK on this board, I guess they will edit my post.)