I would imagine that by the time you can purchase a ticket for a flight an aircraft is already assigned (a year or so in advance). From my own personal experience, when I booked my upcoming YYZ-LHR F trip, flights initially weren't scheduled for the days I actually wanted to travel. Then about 8 months out flights were added. When I booked it 7 months out and went into MMB, I could see that I was on a 14 seat F 777 ex YYZ and a 17 seat F ex LHR. Knowing that there are only 3 17 seat F 777s in the fleet, I imagine that the tail number is already known.

IIRC there was also a post early this year about when G-STBA (the new 777-336) was going into service. Member JumboDriver posted the first fiew days of its schedule.

Now, the big question is, how can we find out without being a BA pilot or cc?