Moderator, Iberia Airlines, Airport Lounges, and Ambassador, British Airways Executive Club
Join Date: Feb 2010
Programs: BA Lifetime Gold; Flying Blue Life Platinum; LH Sen.; Hilton Diamond; Kemal Kebabs Prized Customer
Posts: 63,791
That tube is indeed timetabled to get to T5 at 06:39, and so you should be at a check-in counter by 06:43 with any luck - take the lifts not the escalator.
The check-in time deadline is hard-wired at 45 minutes before the flight, there may be a bit of a queue at check-in, but it's not normally a problem at T5 away from school holiday weekends. So you have a bit of contingency - 20 minutes or so -but not much. If you print or App your boarding pass that will allow you to go straight to check-in. You would have a further 10 minutes if you went hand baggage only - the cut off for presenting yourself at security is 35 minutes.
The Heathrow Express does run earlier, and there are other low cost options to get to T5 before that involving buses, night buses and going via rail stations such as Feltham.