There is an easier solution. Go to the police. There is already case history here and case law. This is theft if they have spent the money.
If the police say it’s a civil matter, then make a formal complaint to the chief constable.
Theft is defined as the dishonest appropriation of property, belonging to another, with the intention of permanently depriving.
Offering to repay £75 per month would not defeat the allegation of permanently depriving if they have already spent the money or even part of the money because it seems unlikely that they could maintain that they genuinely thought this windfall of 10 grand from HMRC belonged to them! There has already been a criminal prosecution for someone who took a similar amount of money from his bank account and spent it knowing that it was a mistake. He actually got a custodial sentence..
Unfortunately, if they haven’t spent the money and it still in their bank account, then you are faced with making an application to court for a court order to get the money back. The police will not be interested.
A solicitors letter will focus their mind more threatening legal proceedings and substantial court costs. I cannot see how they could have a defence to this.
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The thread remains open.
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The good thing about the police is of course that their free although you may need to push them to do something. Even if they just give the culprit a visit, it might be enough to spur the culprit into refunding the money. Certainly, if they were asking the question on here I would be telling the recipient of the money to get it paid as soon as possible because there is a real threat of criminal proceedings against them