Our neighbours had three conifers growing next to our garage. The trunks pushed the roof to one side, damaging the brickwork. I told the neighbours several times what was happening but they did nothing. We don’t want to fall out with them, but we would like the garage repaired. They’ve now cut the trees down.
Ideally your buildings insurance will cover the damage, but it’s possible that outbuildings will be excluded. Check your policy. The insurers would have the option of attempting to recover the cost of the repairs from your neighbours. If you’re not covered you will have to get estimates for the repairs and discuss the matter with your neighbours. But it will be difficult for you to remain friends if you end up having to take them to court.