http://www.gardenlaw.co.uk/
The above site might be useful.
It sounds as though you are responsible for the fence on that side, thought you should know and or check. I say that because you say the last owner erected the fence on his side of yours, and it sounds likely. There is actually no responsibility to provide any fence, but you have done that, and you are happy with it, so end of if you ask me.
If the fence is no good for them, then I suggest they repair their old one, or, probably better for relations, go joint with you on a new one.