Billy,
We recently did the same route with 2 dogs, I second what an earlier poster said, "it won't be cheap!".
There are no direct flights from McCarran until the temps drop - October ish! Arranging transfers yourself, especially if the pets are unaccompanied can be a nightmare.
Ours flew to LHR via Newark with Continental and were well looked after. We used a company called petcarrier.com, a lady called Janice arranged everything, very efficient.
Top tips - Use the Nellis Vets to sort paperwork, saves a fortune! You will still need all that LOA though, well would have done before all the cuts! (cost us close to $3k in the end)
Good luck