They don't really use "Rush" tags on baggage in Europe as much of the baggage handling is entirely automated and containerised, especially if you're flying BA, so there's no-one to read them.
You will need to go through UK Immigration on arrival at Heathrow, then baggage reclaim, then make your way to Gatwick, then through the normal check-in and security procedures. Fast-track security, when it is available, depends on the cabin class you're flying in, you can't buy your faster way through (unlike some other UK airports) or persuade the check-in staff to give you a pass.
Frankly I wouldn't ever have booked this, since there are non-stop flights from LHR to LAS. Could you get an earlier flight from Copenhagen, or transfer to an LHR-LAS flight? You're going to struggle with this, I'm afraid.