Over on TripAdvisor there are a lot of votes for the Feltham-Clapham Junction-Gatwick route.
- Use bus 285 from Heathrow to Feltham.
- Train from Feltham to Clapham Junction.
- Change at CJ for train to Gatwick.

Seems to average 90 minutes end to end.

Cost is £8.50 + £4.90
Buy an all zone travel card at LHR for £8.50 (off peak) and this covers the section of the journey from LHR to South Croydon.
A separate rail ticket is required for South Croydon to Gatwick - £4.90.
You can buy that rail ticket at Feltham or South Croydon - ask for a boundary zone 6 to Gatwick ticket - you do not have to get off at South Croydon.