From the FAQ section of BA's US site:
How can I get upgraded?
Although many passengers ask for upgrades, we must decline, even if seats in the forward cabins are available. Not only would we be unable to accommodate everyone who makes such a request, but also any selection process would inevitably lead to disappointment for others. Additionally, our past experience has been that full-fare passengers tend to learn when others have been upgraded at no additional cost and complain. We also believe that upgrades on a regular basis can damage the credibility of our premium product.
Some further points on this popular topic:
Free upgrades, without mileage redemption, are not among the features of the BA Executive Club program.
Occasionally, circumstances require that we do upgrade some passengers, usually because of an aircraft substitution or an overbooking situation that would leave people stranded.
If upgrades are required, preference is given first to our Gold Tier Executive Club members, followed by anyone who has reached the Silver level.
Please note that there may be times when our airport staff must make such decisions quickly and at random, in order to avoid last-minute confusion or a delayed departure.