BA, 1920 ex LHR? So you're off to Manchester then? ;-)
BA close their check-in desks 45 minutes minutes before flight departure. It follows that they think that it's reasonable to get from the check-in desk to the departure gate in under 45 minutes. So if, having checked-in online, you're actually in the security queue 45 minutes before take-off, you ought to be able to make the flight. However, it would obviously be better to aim to be there slightly earlier than that.
It might be worth spending a little extra and using National Rail services, rather than the Underground.
Heathrow Express departs from Paddington at 10, 25, 40 and 55 minutes past each hour, with a journey time of 15 minutes. The single fare is �14.50 online, �15.50 at a ticket office (or machine) or �17.50 if bought on the train. (Return fares are �28.00, �29.00 and �31.00 respectively):
http://www.heathrowexpress.com/
Heathrow Connect offers a good compromise between the speed of Heathrow Express and the economy of the Underground. Services depart from Paddington at 03 and 33 minutes past each hour, with a journey time of 28 minutes. The fare is �6.90 single or �12.90 return:
http://www.heathrowconnect.com
Chris