Indeed this is a simple ticketing mistake. If you do a search in this forum, about twice a year someone has this problem. If you are at LHR or LGW the agents will just laugh and check the bags in anyway. Menzies at LCY will take what the screen says at face value. There are no CE fares without checked luggage, there are circumstances - often prompted by changes to tickets - where keywords drop out, and with it the baggage allowance. On ba.com/baggage there is a bag allowance calculator and it will show 2 bags for CE (minimum) x 32 kgs and a drop down for Basic fares in EuroTraveller with no bags.

Because BA are aware of this glitch I wouldn't expect any problem with repaying it, so long as the user contacted
ba.com/complaints
There is a customer relations backlog, so if this submission took the form of detailed complaint that will probably take up to 2 months for a reply. If this was just a claim for incorrect baggage, without too many words on the webform, that should be paid much more quickly, 2 weeks or so. There isn't a reliable way of speeding this up. The OP did the right thing by paying up, but it is a training issue at Menzies unfortunately, combined with the ticketing issue.