As you live in SW London, you could try the South West Trains service from Waterloo/Vauxhall/Clapham Junction to
Feltham, from where there is a regular bus to Heathrow. It's not a premium service, so you can use your travelcard/YP card. It shouldn't cost much more than £4 (one way) in total.
There are regular trains from London to Feltham (in Zone 6), starting pretty early in the morning. At Feltham, you catch the #285 bus from right in front of the station. The bus goes every 10 mins or so, and even runs through the night. It is very clearly signed.
The only downside is that it's not great for people with loads of luggage, 'cos it's just a regular London bus.
Have a look at
www.tfl.gov.uk for more details on timings.