I have re-read the thread and the OP indicates that this is a 'double sided' fence. By that I presume that there is one set of posts along the boundary line and to those are attached panels or boards to either side of the posts. So your contention might well be correct here in that each property owns the panels on their respective sides. If this is in fact the case then the willow attachment can stay so long as it is attached to the 'mum's" side of the panels.
But consider a more typical fence of a single panel design, how can anyone logically say that two owners can own those panels half each and separately ? How could any repalcement be done to one half without effecting the other. Does B&Q stock fence panels that have been shaved in half for this purpose? What about a post and wire fence? That just wouldn't make any sense.
I've searched all over for a legal definition of mesne when used in the context of fences and no one seems to want to stick their necks out and give a definition of ownership. Perhaps you have are reference for me?