I don't think there is any quick and easy answer to your question (law is often like that, which is why professional lawyers can make a living), but there are a few useful points to be made.
Although it is difficult to prove a negative, I don't think there is any specific crime or cause of action here. The copyright legislation summary makes no mention of falsely claiming copyright (though bear in mind that the legislation is in many different places, and even the Intellectual Property Office says "we cannot guarantee the accuracy" of their summary); and the fact that the only definition of "Copyfraud" is in a Wikipedia article about an emotive word coined by an American professor shows that this is not a term recognised by the law, which would be very likely if it were explicitly proscribed.
Whether the conduct you mean falls under any other statute is something you would have to ask a lawyer; despite the evident opinion of the Wikipedia author, 'demanding payment you know not to be due' is not the same as 'asserting a right that on careful investigation is not justified', and each case will turn on its facts. However:
- I have spent more hours than I care to remember explaining that 'threatening somebody with a lawsuit' is neither illegal nor tortious, whatever you may think of the morality. If somebody claims that you owe them money and you deny it, the way to decide the point is a court case; unless they take you to court, you need pay them nothing. So what you might call 'threats of legal action' they would call 'explaining the reality of the legal system'.
- If the claimant does take you to court, there will undoubtedly be legal fees; but in the UK, unlike the US, legal fees are paid by the loser. Since part of the definition of "Copyfraud" is that the claimant plainly has no right to this money, the court will find against him and order him to pay your costs. If it is an egregious case, the costs will be ordered "on the indemnity basis", which means that instead of paying any costs you can prove to be reasonable, the claimant will have to cover any costs you incurred unless he can prove them unreasonable.
- It is possible that the CPS could prosecute this as extortion, or the Department of Trade call it "unreasonable restraint of trade"; but before they would even consider either, there would need to be proof that the claimant knew he was not entitled to royalties. If there hasn't been a court case, nobody knows for certain.