Some new neighbors have planted some hornbeam trees adjacent to our existing
Brickwall.
I am concerned that the roots can eventually damage the brickwall, the neighbors
Maintain that the roots will move away from the foundations.
Can anybody help?

Contact the extension service in your county and explain your concerns to them, ask them about the tree

It is possible that roots can and will travel under your wall, I am not sure of that type of tree, many trees have roots that travel 100's of feet. Some go deeper and other stay closer to the surface.

Now on the other hand, he has a legal right on his property to plant any type of tree of bush he wants.

It is possible to havve chemicals put in the soil, or even rock salt in a trench along the fence to keep roots away from it..