Many long-haul Asian and transatlantic flights land early in the morning. Whilst this is not so much a problem during the week, on Sundays I've found myself waiting outside the locked underground station waiting for it to open (with no information to go by and no one around to help), particularly when my flight lands early. To save you some time and worry, here are the details:

First tube - Heathrow T 1,2,3
Mon-Sat 0509
Sun - 0553

First tube - Heathrow T 5
Mon-Sat 0514
Sun - 0558

Note the first Heathrow Express trains are: T5/T1, 2, 3
Mon-Sat 0507/0512
Sun - 0503/0508

Note first T4 departure is 0553 on Sundays
Note the first Heathrow Connect trains are: T4/T1, 2, 3
Mon-Sat 0523/0527
Sun - 0607/0613