I've cycled out of Heathrow. I took the tunnel exit North which is OK as there is a bike lane and then followed the A4. There is a bike lane on the pavement so you don't have to deal with traffic at all. I turned North as I was going up to Barnet. Coming from Earl's Court I'd probably just go out on Chiswick High Street and pick up the A4 after Brentford. Google has pretty good directions and London has lots of bike lanes