With that early a flight out, I would be tempted to get to LHR the night before and stay at one of the local hotels or B&Bs. There is a bus that connects nearby hotels with the airport - the charge is not terribly high, or wasn't several years ago. I stayed at a Comfort Hotel which was around $90 (this was several years ago!); it was ok and geared to late night-early AM travelers.

Enjoy the UK; I'll be there in December!

Bye, Jo in sunny Sunnyvale