Not a BA flight, but I flew to Doha with Qatar Airways on the night that Gulf War 2 started. Five of us in an A330 (c.300 seats), of which only 2 were in Economy. Quite surreal.

My quietest BA flight was a return trip from Hong Kong in November 2006 - there were 2 of us in WTP.

I think back in the days of the London - Scotland shuttle services BA once or twice ran a flight with a single passenger?