I've flown Heathrow - Edinburgh or Birmingham - Edinburgh loads in the past 6-months and every time (and on every airline) I've had to show photo ID, not only at check-in, but also at the gate when boarding.
Might be a good idea to phone BA to make sure they'll accept your US driver's licence as ID (I can't see why they wouldn't, since it is a valid photo ID - but just to be safe...)