Oh bother that makes an otherwise easy job, more difficult.
Yes those roots need cutting off at source e.g. you could make a thin, narrow trench along the side nearest to your neighbour. (no need to remove the actual roots or to dig the soil apart from making that trench)
The trees will be set back a bit but its your garden!
Also they will reroot into your beds so in two years I would do it again.
Best of luck with it Asif.
