Neighbour's garage roof in need of repair and now causing damage to ours

[RoofHelp]

We have an L shaped garage adjoining, alongside and behind, a neighbour's garage.
We had our garage roof fully replaced about 10 years ago and it's still in good repair.
Our neighbour's roof has never had anything done to it in the 30 years we've lived here so it's in bad need of attention.
We now have a serious leak that is where our garage joins the back of theirs. In the recent heavy rain water is now dripping from the sodden boards.
We've had two roofers out and both have confirmed the damage is coming from them. They cannot patch our bit and overlap theirs as there is nothing substantial to lay any repair on.
We have spoken to one householder who said we would need to speak to her wife. So we asked she pop around sometime to discuss. The wife is now avoiding us as much as she can. She's never been one to speak to us, so when we finally do get hold of her, I'm not expecting them to agree to fix their roof or will most likely say they can't afford it.
Anyone got any ideas of how we tackle this?
If they can't afford it, what do we do, as it's the only way to stop our roof being damaged even more.

We had this.
We contacted out house insurance and they sent a surveyor round. Presumably they contacted their insurance because it was then sorted really quickly. We never heard anything about it from any insurance or anything, the new roof was just being finished when I came home from work one day.

Thank you. That's really helpful.
I'll have a look at our insurance.

But if their problems are due to wear and tear then their insurance won't cover their repairs.

I've called our insurers but they're closed now for the weekend so I'll try again on Monday.

I know. And for all we know they may not even have insurance.
They can't actually get into their garage at the moment as the door is broken. So it's all really run down.

They won’t want to be liable for op’s insurer claiming from them for any repairs op needs though.
Maybe it was a case of our neighbours insurance stipulating that they needed to make repairs, I don’t know, we never discussed the roof issue with them again as it was sorted. It was just all far less of a worry than we I I thought it was going to be.

I've spoken to insurance company this morning and they're sending a surveyor around to have a look.
So I have everything crossed it gets sorted as easily as yours did @DoIOrNot