Bus 285 from HGI to Heathrow Central (T2/T3) is free. I won't take the tube on this route because of the time I missed a flight due to being stuck underground between stations for 20 minutes.

Buses 482 and 490 run between Hatton Cross (3 mins walk from the HGI), T4 and T5 and are free. Annoyingly in the very early morning, they both run at 30 minute intervals but both buses arrive at roughly the same time instead of spreading the headway. They also tend to be completely packed with airport workers, and travel time is about 25 minutes. For the train it would also be wise to check timetables.