You have a claim against the neighbour in common law negligence.
It will help if I explain how claims in negligence arise.
There needs to be
1. a duty of care
2. the duty of care needs to be breached
3. as a result of the breach there needs to be loss or injury
4. the loss or injury resulting from the breach must be reasonably foreseeable.
In order for there to be a claim in negligence, all 4 heads have to be satisfied.
In these circumstances in my opinion you have a claim.
You can’t beat the neighbour with a blunt instrument until they pay up and if they don’t, your only remedy is taken to court. I would start by getting a quotation for fixing everything and telling them that’s what you want. If they don’t pay up when your deadline expires tell them that you are going to take them to the County Court and this is their last chance.
The new Moneyclaim beta system is easy to use: https://www.gov.uk/make-money-claim
the new system is much better than the old government Gateway system.
You can prepare the proceedings now, at your leisure, save them, and then just pay the fee and press the button when the deadline in your letter before action, expires.
It's then just a case of following the court process and responding to any emails that you receive.
Thank you for letting me assist you with your legal question. I am glad that I was able to help.
I am not certain whether that answers the question for you or not, but I am happy to answer any specific points arising from this.
It will be my pleasure to help you again either further with this or any future questions you have
Kind regards
Stuart