Off peak (after 09:30) fare from Gatwick Airport to Heathrow is £18.80 using Oyster or bank card.
Change trains at Farringdon and Paddington.
Gatwick to Feltham (change at Clapham Junction) is £9.00 plus bus £1.60 = £10.60
Fares info here:
https://tfl.gov.uk/fares/find-fares/...r?intcmp=54716