Q.Lilac Tree Root Cut
My neighbour has done some work in her garden which has obviously cut roots to well established tree – will it grow back

Certified GKH Gardening Expert
It depends on how much roots were cut, and how big the tree is. Usually, cutting a few roots, here and there, is no big deal.
If the tree is large, then there was likely no harm done. If the tree is established, but small in stature, then damage may have been done. However, I'm not familiar with laws in your area regarding things like this. In my area of the US, Anything may be removed on your side of the property line, as long as the tree does not die.
This one sounds like an issue to be resolved by your local law enforcement or council.