The answer to your question will depend upon:
(a) how far you want to drive ; and
(b) whether you've got the time (or desire) to see somewhere else on your travels.
The nearest airport served directly from the UK is Los Angeles International. You can fly there from Heathrow with Air New Zealand, American Airlines, BA, United Airlines or Virgin Atlantic. (Norwegian Air Shuttle also fly from Gatwick). You'll then have a 2-hour drive to Palm Desert.
Alternatively you could fly to one of the airports shown here
http://psp.fltmaps.com/en
and then take a flight to Palm Springs Airport, leaving you just a 25-minute drive to Palm Desert. (That would, for example, give you some time to explore New York, if you so chose).