A couple of other options (sorry, you'll have to forgive me being geeky about this, I work for the railways in my day job...):
1. TfL 490 bus from T5 (or 285 bus from Central Bus Station) to Feltham, train to Clapham Junction, then connect into a Southern service to Gatwick. All of those connections run frequently, rail cost would be £18.30 one-way plus £1.55 (?) for the bus. I've done this a few times in the past and I reckon with a following wind, if all the connections are smooth, you could make it in around 90 minutes. The bus stop at Feltham station is right next to the station, with step-free access to the platform. Clapham Junction is a big place, but it's well provided with lifts if you use the overbridge (not the subway!) to change between platforms, so it's not too bad even if you have luggage.
2. Probably more a theoretical option than one that would be useful in practice, but you could take the RA2 RailAir bus to Guildford and change there onto a GWR train through to Gatwick. The main caveat with this is that both services are only hourly, and I'm not sure what the connection times are like. It would probably be much more logical just to get a through NX bus straight there.

EDIT: Firstly apologies for duplicating what was already above about the route via Feltham. The fare I've quoted is for a paper ticket from Feltham to Gatwick. It's significantly cheaper to use contactless payment or an Oyster card - this would then be £12.90 peak (0630 - 0930 and 1600 - 1900 Mo-Fr) and £8.70 outside these times. Bargain! Please see also my post below about the change in location of the bus stops at Feltham station.