A few years ago, the CVV falloff thing happened to us. Never got the "call us" email. But noticed I couldn't do online checkin. When I checked-in at the airport was told there was a problem. After 5-10 minutes told me the upgrade hadn't been paid for because of no CVV. Asked for my credit card, took the payment, and all was well, reserved seats intact.
It is a bit surprising that BA must be aware of the problem, but not able or willing to fix it.