beech trees under stress

Beech trees 60 feet tall may well have been de-stabilised by cutting their root plate within two metres of the tree trunks. If the trees are on your property, they are yours, but their roots have travelled onto adjoining property, and the owner is within his rights to cut the roots, even if as a consequence it makes the trees more likely to fall. Although by losing roots on one side, they are more likely to fall away from the side where the roots were cut.

Cutting 60 feet beech trees down to 15 feet will spoil the trees and leave very ugly stumps for many years, until they re-sprout, and even they will look bad when the leaves are off. When they re-grow, beech tree that have been cut hard will develop poorly attached branches that can fall.  

Probably the best solution is to take down the trees completely and replant with  new trees, possibly smaller in eventual size. It might be possible to plant these at a semi-mature size to speed up the restoration. The neighbour might be minded to pay for all this in order to have the trees removed.

Whether the trees could come down on the neighbouring house depends on how far away they are. If they are 60 feet tall, they need to be 60 feet away minimum.