Hello! I've started thinking about this myself, since I'd like to take Chloe, our Cavalier King Charles Spaniel, to the U.K., but I decline to ship her under the plane. My best thought so far -- and I know it will work, but I haven't yet put together all the details -- is to fly with her in-cabin to Paris (we're going on United this fall, and several other airlines allow small dogs and cat to travel in-cabin to France), then rent a car and drive to the U.K. via the Chunnel. You'd have to meet France's import requirements (not hard, your vet will help), make an appointment with a Parisian vet (do a search on my website Dog Jaunt for "English-speaking veterinarian in Paris," and you'll find one that people like), get a E.U. "pet passport" from him (again, not hard, and you'll have pretty much done all the work already to get into France) and confirm that you've met the U.K.'s import requirements (not hard, and again, you'll already have done the work to get into France), rent a car and make a reservation on the Eurotunnel Shuttle:
The Channel Tunnel - Drive Between the UK and Europe Through the Channel Tunnel - Eurotunnel Facts and Information
It's a little roundabout, but it would really improve your dog's travel experience, PLUS you'd get at least a day in Paris. Nothing wrong with that!
Here's my website, which also has (under "Guides") a quick reference chart for international airlines -- which ones allow in-cabin pets, and an overview of their policies. It's called Dog Jaunt --
Dog Jaunt : Traveling with a small dog
I wish you and Ozzie a safe and easy trip, and a great new life in the U.K.!!