FYI--the Thursday email is not just confirmation of pre-planned work. Occasionally, last minute work has to get done.
Yes, there is replacement bus service but that will throw your schedule out of whack.
At 4 AM, do you really want to start hassling with the tube? Walking to the station? Possibly dealing with a bunch of drunks returning home from a night of partying?
Arrange for a car service, they pick you up at your hotel and drop you off in front of the terminal. It's cheaper than a taxi.
Or you could take a chance with Uber. Also probably cheaper than a taxi.
I have taken early flights out of Heathrow. I stay at an airport hotel the night before just to make my life easier. And get an extra hour of sleep. But you may not have that option.
Nigel posted as I was writing this. I agree with him. And this from someone who always take the tube to Heathrow--although not that early in the morning.