To answer your last question first, probably the only way to save your hedge is not to move it. The plants that make up a hedge intertwine with each other to such an extent that the only way to move it would be to cut the whole thing down drastically and then move the individual plants and shrubs that you are left with.
It is likely that the hedge will recover, but of course it will have to grow from scratch all over again. The bird habitat would be lost although again they might well return as the hedge establishes itself, which is likely to take five to ten years.
What’s Over the Hedge?
Let’s look at some of the other issues surrounding this, both gardening and non-gardening. Firstly; it is not necessarily the case that the hedge is responsible for the barren conditions on the other side of the fence. Without knowing the main plants in your hedge and the plants that your neighbours have tried to grow it’s not possible to say this for certain though.
Do plants grow at the base of the hedge on your side? If so, then your neighbours are probably trying to grow the wrong plants. You could take a look at the Royal Horticultural Society website and see if you can suggest some plants that are more likely to grow in the conditions they find on their side of the hedge.
Secondly the non-gardening issue. Your neighbours have no right to ask you to move your hedge. What you do on your side of the fence is your business and what they do on their side is their business.
Hedging Your Bets
Of course for the sake of good relations with your neighbours you have to temper their rights against reasonable requests and negotiate with them. If they really can’t grow anything on their side then suggest that you help them to put a barrier in below the fence on their side.
Dig a trench about two feet deep and slot in something impermeable such as old paving slabs (place vertically) or a polythene liner which you can peg in place initially. The weight of the replaced soil should hold it in place then the pegs can be removed.
If what they say is true then you will find lots of hedge roots as you dig and you will have to cut through these. This will hurt your hedge but it should recover, and then if your neighbours still can’t grow anything then they can’t blame it on your hedge.