Digging up an old thread here as I have experienced the same issue this week.

Called on Monday to make a booking for about $6k. Booking was made but not without an error by the sales rep who apparently entered the expiry date of my card incorrectly. Finally the transaction went through but subsequent to that my card was declined a couple of times. I called the credit card company who advised that BA had placed a pending charge of $6k in addition to the actual charge. It is scheduled to drop in just over a week's time but until then I can't use my card as it would put me over the limit.

The credit card company advised that the only way to remove it was for BA to send them a fax.

Well, I called BA. The first guy said he'd speak to his manager and he put me on hold for 40 minutes, at which point I guess the 'hold' timed out and I went back to the main menu. The second guy put me on hold for about ten minutes and he told me there's nothing he can do because, and I quote "BA does not have the ability to send a fax in this department" and that I will have to wait a week for the pending charge to expire. I asked him what he suggest I do if I need my credit card in the meantime, to which he said he doesn't know what to say. I'm currently on hold again as I asked to speak to someone else. 1 hour and 7 minutes in total so far, and counting...

Bit of a rant but just wondering if anyone's had any success with getting these pending charges removed?