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You have two choices. Either you get your neighbour to pay the money themself which is unlikely, or you take your neighbour to Court. In reality this will not happen as your neighbour will pass proceedings on to NIG.
If he refuses to pay, just have a word and say I have to issue proceedings against you (as it is his responsibility) but you should just forward these to your insurance company. You should include all losses that you have incurred as a result.
I am sure the insurance company will end up paying out very quickly.
You do not need to provide beyond doubt in the County Court - only balance of probabilities. Was it more likely that not.