If you booked normal tickets then I doubt you would have a chance of an upgrade in all honesty, but then I don't fly BA that often. I was upgraded a few times on BA when I worked for another OneWorld carrier and traveled on duty tickets, although only within Europe, but my priority code specifically allowed upgrading on a space available basis.
I'd say you would probably have more chance of an upgrade if you were using your staff standbys, at the very least you would be getting the same flight for cheaper and with more flexibility if you used an ID ticket.