If the actual location ends up being somewhere South East-ish of the airport, I wonder if the least hassle, relatively robust (on timings), albeit on a good day probably not the absolute quickest option is Gatwick Express >> District Line >> Picadilly Line. And exit somewhere like Hatton Cross.

Just a thought. As I said, probably not the quickest, but then again I'd have thought less chance of a catastrophic delay (which can definitely happen if you get a pileup on the M25 while en route).