The new Amex Centurion Lounge in Terminal D has shower facilities and a nice buffet. It is convenient to AA, DL and UA flights.

If you're arriving on an international flight and connecting to a domestic flight, I think UA would be your best bet, since you'll arrive in T3 and UA check-in is in the same terminal. I'm not sure of the procedure if you'd be departing from an airline with check-in counters in T1, like AA and DL. Presumably your bag will be tagged all the way through to LAX, as I think BA/VS (whoever you're flying in from the UK) interline with all of UA/DL/AA. If so, there is probably a bag drop right after customs, but I'm not sure if there will be an AA/DL agent to print your boarding pass. However, you might be able to do OLCI before your UK departure and print your BP, and the kiosks in LAS which are often generic (multi-airline) might be able to print it as well. You'll need it to get through security before you can get to the departure gates.

Thanks for the tip, my connection is now the next mornin...the turn around time from Landing to Immigration, terminal changes etc was way too tight to catch the domestic :)