Very definitely a vishing attack. There has clearly been a data breach so needs reporting to BA’s DPO, but the breach could equally have been from your emails so make sure you change passwords to your mail accounts as well. All of the information they had sounds like it could have come from emails sent to you by BA not necessarily from BA’s internal systems so it could have been you rather than BA that was hacked.
I know this is a little locking the stable door, but never proactively provide personal information in a case such as this, get them to confirm the information to you from their records. I had an attempted vishing attack trying to get me to provide details for them to handle compensation for a car accident. I already knew it was dodgy so asked them to confirm if they got the details from Churchill, my insurers, and that this was the accident in Hull. When they did, I informed them that I wasn’t insured by Churchill and had never been to Hull at which point the line suddenly went dead…