This is part of the structure of the properties and it is classed as a party structure and hence, you are jointly responsible for repairing.
If they want it taken down because they no longer use it and you want it put backup because you do, and if this came an argument in court, the court are likely to maintain the status quo and require them to pay a portion of the repair, probably have an element of proportionality according to use.
Hence, what’s likely to happen if it came down to an argument is that you would be required to pay perhaps three quarters of the cost of the work and they would be required to pay one quarter.
There is no exact formula for this and I am just giving my opinion.
If the chimney stack is dangerous, the local authority and serve a dangerous structure notice and get a court order to make you have the work done. The neighbour will get a similar notice. The dangerous structure doesn’t have to be dangerous so that passers-by would get injured if it collapsed. Would be just as applicable if it was a chimney in the back garden not affect anyone other than the house in question.
Can I clarify anything for you?
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