Google Maps has a pretty good knowledge of the London public transport. This map will show you several options, which will all have you arriving before 6:40am on Saturday (tomorrow--although any Saturday ought to be roughly the same).
The one that looks the most attractive to me is the last one, which is to take the 295 bus (although 283 may be just as good) to Hammersmith, then the Picadilly line to LHR. It includes only 6 minutes of walking.
On the London Tube web site you can see the current status as well as planned outages. Whether a strike-caused outage would be posted early enough to help you make your travel plans, I have no idea. I suggest waking up a bit early and checking, just in case. Then at least you'll have the option to call a taxi in time, if the worst should happen.