It would have perhaps been better to cut the ivy back as soon as it started climbing up and over your shed. We have to prune our own C. montana ruthlessly or it would be all through the inside of our potting shed. As to whether you can claim on insurance I honestly don't know.

I would have thought that a eucaliptus that close to your home was asking for trouble. Suggest you talk to the local council about the problem if you aren't able to talk to the neighbour.

Thanks moon_grower
The ivy this year has grown like mad , I was ill all last year so the very basics got done, just put 2 ,3x3 on shed roof to stop anymore damage to roof felt.

It would be good to replace the roof felt before the winter, but check with your insurers first in case you are covered and they want to come and look at it. Also check your policy to see what your excess is - it may not be worth claiming.

You are legally entitled to cut off any branches that overhang your garden (you are supposed to give them back to the owner but this can look like nastiness - I know as it happened to me with a rose once) It is wise to mention first to the neighbour that this is what you are planning to do. You are right though,as MG says, it is very close to your house and perhaps you could mention this at the same time, though you might get a dusty answer. So many people plant trees and shrubs very close the their fence and it is always a recipe for trouble later.