You could give the neighbours permission to cut the hedge?

Adding a comment so that I can read all the answers..
and I've included this in GoYpedia Screening Ideas. :o)

To remove this is a huge job and you might find its not that easy to redefine the boundary line? As some of it might well be rooted in the neighbours garden by now? I would look again at getting a garden maintenance firm on a yearly contract to look after it. I have heard you can spray with growth retardant?

I agree with Drc. I have a high hedge too. A retired neighbour of mine cuts it for me every year for a reasonable charge. He is pleased to earn a bit of extra cash. You might be able to find someone like that. I think you would really miss the hedge, which looks lovely, but if you do go to the trouble and expense of removing it and putting in fencing, you will certainly need to liaise with your neighbour about the ivy! I do sympathise as I seem to be getting to a point myself, where certain aspects of the garden are getting difficult to manage.

Thank you all for your suggestions.

Bulbaholic : the neighbour on one side seems to like a jungle atmosphere and the other neighbour hardly lives there - only comes to cut his grass and hedge. The hedge is entirely in our property.

Garden maintenance firms in Brussels, Drc726, are very expensive - it costs 250 pounds sterling just to cut the hedge!

In the interim, we think we will reduce the height and see how we get on.

That is a great deal to spend twice a year Avril. But I am not convinced you would like fencing all the way down? But if it comes to it I dont see an easier alternative?
One other idea is to partition the back 2/3rds off with a trellis and just cut keep the top 1/3rd as hedge near the house and fence the rest?

First can I say what a lovely garden and the hedges make it so private and a good backdrop for the planting. I think the idea from drc is a good one, a gradual move towards fencing would be what I would go for, slowly getting used to the difference and less cost as the hedge reduces!

Gosh that is one thick hedge, it looks to take up quite a lot of your garden and light. If you take it out, you will be left with some sort of no man's land and will not only need to re fence but also create either lawn to the edge or a wider border. I think this will also create more work for you.

Perhaps taking down the height and thinning it out might be an option. You would certainly have more of an open feel about your garden and not feel so enclosed.

My nan had her hedge taken out 7 or 8 years ago, when she couldn't cope any more. I think she regrets it though, even though its less work. To be fair though, the people who put it in were idiots, its about as straight and level as a dogs hind leg. She now has door stops wedged in to the pannels to stop the rattling in the wind.

I think Drc's idea of slowly moving to fence (if you decide to do that) is a good one. Gives you change to get used to it, and if you find you prefer the hedge you can keep it part hedge for the aesthetics.

In the meantime are there any young lads (or girls) living round by you who would be willing to cut it for you. There are a few round us who could probably be persuaded to cut the hedge occasionally for a few pound. My brother and cousin help my nan sort the hedge in the front and cut her lawn. I get dead plant removal duty lol.

Stunning garden by the way.