I agree with you - the concept of "privity of contract" applies in this situation. Your friend had no contract with the customer, you did. So you should have been consulted by the customer and as you point out, further work was carried out outside of the contract. I do not think you are liable for this - the consumer law is clear in that if the customer has issues with the quality of the work then the company which installed the flooring needs to be given an opportunity to carry out remedial works. There is no situation under the Consumer Rights Act 2015 where it would make you liable if your friend carried out work and so on. You should refuse any payment and defend a claim if one is made.
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 (and it wouldn't in this case) 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.
I can assist you going forwards if it gets to the point they issue a claim. 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 : https://stormcatcher.co.uk/practice-areas/dispute-resolution/small-claims-lawyer-service/