Neighbours tree.

It’s your responsibility to report it and make a claim through your insurer. The neighbour may be legally liable if the branch falls and causes damage as a result of their neglect in taking care of the tree, such as if dead or rotting branch falls and you had previous knowledge of its unhealthy condition. Although you had every right to cut down the branch if it was over hanging on your property, to prevent this from happening.

You are entitled to claim back the cost of repairs. And also entitled to cut branches back to the border line.

If the tree comes onto your property you have rights to cut it back to stop It effecting you damage should be discussed between you however a claim would be next step

Your first mistake was offering to go halves on it. As none of this is your fault. However, this will put you in a good position to claim off of them. I would do this without consulting with them again.

Your aloud to cut back the tree to your border but if it’s damaged your shed roof they have the right to pay for the Damage that’s been done maybe have a calm chat to them to sort the problem out