BA don't really do standby I am afraid, hence the answer you were given. You would either get booked on to an alternative flight if space existed, or if it is full you would be rebooked on to another flight.

EDIT: To add I did manage to get on am earlier flight once when I was affected by a cancellation, but that involved me asking the gate staff and hanging around the gate to see if there were any no shows since the flight was otherwise full. There is no formal standby, and there is no list the call centre can add you to.