I don't think you can get access to the raw number of unsold seats - commercially confidential information - without inside help.
What is easier to get is the displays that travel agents see, showing what is available for booking. Full-ish flights will be identifiable by restricted availability in certain or most classes. For Internet links (thanks to other FT'ers), try:-
http://www.aeroplan.de/ap_rb_galileo.htm
(or
http://cyber.law.harvard.edu/people/...leo/frame.html for an English language interface)
http://www.etour.co.jp/Solar/index.html
(or the link marked "switch to Japanese Etour interface" at the top of
http://cyber.law.harvard.edu/people/...leo/frame.html for an English-language interface)
http://flyaow.com/classual.htm
http://flyaow.com/classamex.htm
To see what seats are available for pre-allocation, the easiest way is to make a dummy booking on Travelocity. Pretend to make a booking on the flights on which you are travelling. Go beyond the screen where you enter your name and personal details. If seat pre-allocation is available, a seat chart should come up for each sector in turn showing what can be pre-allocated. Armed with this information, call the airline and ask for a pre-allocated seat that you like. This can often also give you an idea about which of two flights is more full.
You should note that BA will only pre-allocate about 30% of the seats before check-in opens. If there is nothing there that you like, register a preference. Try checking in early, perhaps online if you and your itinerary qualify - see the BA website for details. That may get you a better selection of seats than if you wait until you get to the airport. But if you have no status then there will be people ahead of you in priority for the "good" seats.