Hi thedaras,
What follows is mere speculation, because I don't know the property in question or even the plan of the house.
I'm also not a qualified legal person specialising in land law and you may need to take professional advice on this.
Issues at or near boundaries, or to do with shared services or drains can be thorny ones, especially if the gutter, which "comes across onto her side of ... her house" is over your mother's land - it is on her property.
If the leak arises because this is a shared gutting running straight across the backs of the houses and a joint has gone on your mother's side then the liability may rest with your mother for not re-fixing or re-sealing the gutter.
If the joint was broken due to works or a lack of maintenance on the neighbouring property and not re-sealed then the liability may rest with the neighbour.
In other words, depending on how this arose there may be different allocations of liability.
If the damp arises from the gutter becoming blocked partly because the adjoining property holder does not have access to clean the gutter - assuming it is as I suggest above - then the issue becomes how to avoid this recurring going forward, assuming the current problem can be sorted.
Your mother may not want to allow workmen with ladders access her property.
Regardless of liability, whether or not the insurance covers it depends on the contract wording and exclusions in the small print.
I know this isn't a cut and dried answer, but I hope it gives you some idea of the possible complexity of such matters.
ONQ.
[broken link removed]