The Consumer Rights Act 2015 applies. It states that if remedial works are required, then you are entitled to return to attempt one repair. You were not given that opportunity so if the customer were to sue for their money, they would lose the claim in my view. They need to prove you caused damage around the house too - the onus is on them as a claimant to prove you damaged the property in some way. As it stands you should refuse any refund. Unless there is some truth in what they are saying.
If they do sue, you will receive court documents (a response pack) which you must complete and return to the court. It is a tick box exercise for the most part and there is a short section to write a defence which is easy enough though please feel free to come back to this site if you need any more help. A claim will also take 9-12 months to be decided at court. If you lost then you would get 14 days to pay the judgment before the claimant can enforce the order, and 30 days to pay in full before it is registered with credit agencies. The claimant cannot recover legal costs if they win, in a claim under £10K (a small claim), all they can claim are the court fees and interest.
If the claim has no merit then you have an option (after your defence is filed) to apply to strike the claim out. The court will consider an application if the claim has no merit, or is misconceived. The application costs £255 but this is recoverable if your application succeeds.
The hearing (if the case gets that far) is likely to be held remotely, it's you, a district judge (who is a practising solicitor or barrister) and someone from the claimant company. The Judge decides and if you lost, you get 14 days to pay the sum. If paid in full within 30 days then nothing goes on your credit record. But certainly I think you would succeed in a defence, just to be clear, based on the facts in your question.
I can assist you going forwards if it gets to the point they issue a claim though as we are an internet forum (not a law firm), we can only give information and guidance from a legal standpoint.
If you need a law firm to assist you to defend the claim, here is an example for you and they offer a fixed fee :