Yes, it also applies to Scotland.
I have to agree with Zacs. With all respect to Atheist and Goodgoalie, there are no accepted "rules" on fence ownership.
They are all myths that have developed over the years.
I must admit I would just put up my own fence on my side, right up against the existing one.
To be pragmatic, I would do as Dougie says........... go and tell them you're going to do some work on the fence to stabilise it.
Since no one will have a clue as to ownership, I doubt if anyone will contest it.