hippy if you are planning on staying long term I would erect your own fence next to the neighbours. You can paint it/stain it and grow things up it because it is yours.
It makes for a happier life if in the future new neighbours move in and demand you remove plants from their fence.
I would photograph the fence as it is now, but also video it with running commentary. That way you can show what state the fence was in when you took the decision to erect your own fence and why.
Normandy the law allows you to trim back any branches/bushes back to the boundary line and by law you have to offer the cuttings back to the owner. This doesn't ever mean chucking them into their garden 
A boundary line is a very hard thing to define really. On a title plan the scale would mean that genuinely some boundary lines are incredibly thick. They sometimes line up with something but still a fence is nowhere near as thick as a boundary on a title plan (I have spent way too long on GardenLaw website)
So I would tell the neighbour what you intend to do. That way it gives them the opportunity to cut it back themselves. Also make it sound like you are doing them a favour with well I wouldn't want you to end up with lots of balls in your garden from my children etc etc. Sweetens the deal a bit.
You can erect the fence on the boundary but that would mean cutting the hedge back beyond what you are allowed. So if they aren't happy I would just put it on your side of the boundary.
My FIL was unhappy that the 3ft fence between his next door neighbour and the next door but one neighbour was being replaced with a 6ft fence because it would spoil his view of being able to look over lots of gardens and not feel closed in. And that wasn't even his fence with his neighbour!