I've not read anything like the full thread (though I saw what looked like some pretty sensible input from Dumbledore). And don't take this as legal advice. However....
The contractor likely owes a duty of care in the tort of negligence to your neighbour. Just because the contractor didn't contract with your NDN doesn't mean that the contractor cannot be liable to the NDN.
I'd be inclined to write to the contractor, including a copy of any correspondence from the neighbour (or, if none, setting out what the NDN has said) and including copies of the quotes, and pointing out that this is the contractor's responsiiblity to sort. You should then write a letter (so that you have paper trail copies) to NDN saying that you have written to contractor to inform them that they need to sort this out. You should invite them to pursue the contractor directly - make sure your NDN has the contractor's details.
The aim would be to encourage your NDN to pursue this with the contractor, as he / she is likely entitled to.
Document everything in case needed!
If the NDN takes you to court, you might consider either joining the contractor in as an additional party, or considering bringin a claim separately against them for an indemnity. Have a look at the Civil Liability (Contribution) Act 1978 and some online guidance around that, as well as a think about how this represents a breach of your contractor's contract wth you.
You really would benefit from a couple of hours of a lawyer's time. Its not complicated stuff, but a bit of helpful direction at the outset could save a lot of time and hassle at a later stage.
Hope this is of some use...