Okay, so I'll give an update, for future reference about this topic.

The first call to BA was quite rude, "Just wait to be ticketed, could be day before departure." and "There's nothing I can do for you right now".
Second call had more success, managed to get a more proactive agent who with the help of me giving all the flight details and payment details (again) ticketed my flight.

I think for future reference it's good to call BA within 1 week (that is apparently the maximum period that they are allowed to store the relevant credit card details) when not ticketed yet.