To the OP, my suggestion is to do this over the phone. The agent has in my experience always been able to 'grab' the space on the phone, then process the cancellation, then ticket the new booking. This has always been on BA operated flights but i always got the impression they'd in effect start a new booking ('Booking 2') (but in such a way that the seats come out of inventory and are 'yours') then open a new 'tab' to process the cancellation of Booking 1, then switch back to take payment for and ticket Booking 2.
I'd do that to be honest. Apologies if I've missed something.