There isn't any one airline that's cheaper than others. They all have various fees (some quotes include all the costs, some don't and then you have to pay handling fees on the UK side). Quotes depend on the dimensions of the kennel your dog will be in, and the weight of your dog as well. Check the approved airlines for the route(s) you can travel with, and call them for quotes.
You probably won't be able to book your dog on the flight until 2 weeks beforehand, but at least you can get a sense of how much you'll pay now. FWIW, I paid just over $2,200 back in April 2010 to move my 60lb lab mix dog and 12lb DSH cat from Washington Dulles to London Heathrow with British Airways. I think about $500 of that was the cat, so my 60lb dog was ~$1,700. Prices have almost certainly changed since then, but I don't know by how much.