As a long-term, regular and generally supportive user of BA I can honestly say the call centre is BA achille's heel (the T4 baggage reclaim problems now behind them). Many, many times over the years I've found the service and plain usefulness of the call centre 'adequate' to 'useless'. If they could solve this one I'd fly then at every opportunity. As it is, when they p#ss me off, I try someone else for a while.
They're not alone in this respect, but they've had an opportunity to raise their standard against competitor's offerings for a long time and missed the opportunity.