Welcome to Flyertalk and welcome to the BA Forum opeolat, though I need to recognise the poor experience that brought you here.

I've been giving out advice in this forum for a few years now, and it's rare when you come across something completely new, like your case. Mostly people have difficulties in a spectrum of existing or known issues, so it's quite easy to give appropriate advice. But suspending a ticket just before travel, and not being able to explain why is new to me, though it won't give you much comfort. I have heard of cases where banking companies have recalled their payment, such as via Section 75 of the Consumer Credit Act or Cashback. In that situation a consumer would be clearly in a situation of holding a unpaid for ticket, usually the consumer would be aware too, and thus it is resolved by making a new and immediate payment. This does not fit your circumstances in a number of ways.

The other issue, much more common, is that a ticket has not been issued. This is a technical stage which is prone to glitches and which should have been resolved once you were talking to an agent, it normally takes just a few minutes to requeue the ticket for issue. Again the wording you used doesn't seem to relate to this. However there is one area where there can be problems - if the payment is made, but BA thinks it hasn't got a CVV for you, it can show as as unpaid to BA, but paid on your side. Payment did go through without the CVV and BA thinks the CVV is needed before that payment can move through the system.

I can perhaps help on the next step: at some point BA will reach out to you, hopefully in the next few hours and either say "sorry our fault" or "there was a payment problem and it's still not resolved". Once that is done, you will then be offered the choice of a full refund or a rebooking, essentially for dates of your choice. I think it's supposed to be within 2 weeks of original date, but if BA was at fault they may be flexible on that. If it was a BA problem you can also claim Involuntary Denied Boarding compensation of £520, and pay right to care costs (e.g. extra taxi fares, depends a bit on what you did after check-in). You may be able to raise a complaint with DOT about this too, but from what I know IDB in the USA is purely about oversold cabins, which appears not to be the case here.