Then you're best to ask the bank what the appropriate proceedure is. It can't be the first or last time this has happened and they'll have a standard letter to send the recipient.
Had they realised they'd given the wrong details, obviously they would have corrected it immediately - and I know the bank has done nothing wrong. I just wondered what the legal position of the recipient is.
It's equivalent to 'theft by finding' - the recipient knows it is not their money and should return it immediately when asked.
If it's a modest amount of money and the recipient has spent it (perhaps because they are disorganised and haven't checked their bank account in detail) then they will be given some leeway in how quickly the money is reclaimed.
However enforcing this can be problematical - the bank is not allowed (without an expensive court order) to tell you who the recipient is, so taking legal action against them is going to be difficult.
So if the recipient just ignores any informal request the bank make for it back he's home and dry.
This happened to me once about 30 years back. I phoned a solicitor and he said "what do you think?". I said "I suppose it is someone else's money and should return it". He said "There is no morality without pain young man, phone your bank now."
I phoned my bank who deducted the amount that day and promised to return the money to its owner.