Just wonder if it makes sense to take the bus from Heathrow to Woking, then train to Gatwick. Depending on timetables it seems that it could be done in about 2 hours and wouldn't traffic be less on that bus route?

"wouldn't traffic be less on that bus route?"

Absolutely not. The really bad traffic congestion is getting from Heathrow to the M25, then on the M25 as far as the M3, From J11 (the turnoff for Woking) to Gatwick, it's usually (even at rush hour) reasonable.

And the train from Woking almost always means changing at Clapham Junction (or relying on the often delayed train from Guildford): a horrid shlep between platforms by stairs, and a near-nightmare trying to find the platform indicator at either Clapham or Guildford.

Actually, the most reliable route at rush hour - though not one I'd recommend to someone unfamiliar with London and quite pricey - would be:

Get to Heathrow Express from T3: 10 mins
Wait for Heathrow Express: Average 7.5 mins
Heathrow Express to Paddington: 15 mins
Escalator to Bakerloo Line and wait: 7 mins
Bakerloo to Oxford Circus: 7 mins
cross platform to Victoria Line and wait: 5 mins
Victoria Line to Victoria: 5 mins
Get to Gatwick Express and wait: 10 mins
Gatwick Express to Gatwick South: 30 mins

I make that roughly 100 mins - and all these links are, by our standards, exceptionally reliable.

It's what I'd do at rush hour if I didn't have much luggage, but getting to the Gatwick Express at Victoria requires a walk up a flight of stairs many people just can't manage with heavy bags.