At the end of the day, your wife made the error and transferred the money to that person and gave him the money. The bank can’t just go in to that persons account and take the money back.
All they’ll do is send a letter to that person and ask him to transfer it back.
I would still kick off with First Direct and ask to make a complaint. This sort of story has been making the press quite a bit recently and it wouldn’t surprise me if new rules get put in place for it pretty soon.