Your neighbour must be a nasty piece of ****e, for him to do this, a good neighbour would never do that. A good neighbour would know that his gutter is pretty old and in a bad state and will eventually need repairs or replacing with a new one, without looking for an excuse or waiting for an opportunity to snap at and then present you with a nasty and shocking bill by holding you responsible for damage to his crumbling old gutter.
He is taking advantage of your good nature.
You really don't have to prove anything regarding the condition of his gutter. Whether it was good or bad, or damaged, but before he engaged his insurance company or anyone, If he had any issues with you and his gutter, he should have first consulted you, discussed this matter with you, and he should have made his intentions clear to you that if you don't do anything to fix his gutter then he shall get someone to fix it and give you a bill, so did he do that ? If he had any concerns that your builder damaged his gutter, it would have been at this stage that he should have given you the opportunity to address his concern, or fix any damage caused to his gutter, Did he do that? Did he tell you that he will be seeking a claim or componsation?
So only then if there were real issues, and you ignored them or chose not address them, then he has the right to get it fixed and put a claim through.
Further more these sort of things should be in writing as well, so no one can deny.
So write back to his insurance company that you are not liable for anything. as this has come as a complete shock to you and you will not pay a penny or entertain any part of this claim since you were not consulted or knew anything was wrong with his gutter, if there were any issues then you should have been given the opportunity to address them, but you were not consulted by anyone.
Get a solicitor to do that, I know it may cost you a few bob, but you have a £5K claim on you, so if you have to pay a solicitor £250.00 then it may be money well spent. If you really don't want a solicitor, seek free advice from your CAB.
( But that is what I would do, write back, and say you knew nothing of it and your neighbour had not discussed anything issues with you or gave you any opportunity to fix any issues, so you are not going to entertain this claim what so ever. )
Did he come over to you and accused you of damaging his gutter?
or discussed anything that he was not happy with?
then he has no right to present you with a bill nor his insurance company.
Like I said, I think he is devious, he tried to make a claim from his insurers that you damaged his gutters, you will probably find that his insurers will be more sympathetic to you especially if you tell them that you have all the proof in photographs and on google images, they know many people are devious and fraudulently make claims blaming damage on someone else.
I think they will most probably end up taking him to court for a fraudulent claim.