If the worst comes to the worst, a simple fence like fidget is suggesting with ivy on it will give you a 'green fence' and over time you can add other things. A montana clematis would work there - it's just a question of planting a few few feet away and getting some decent soil and manure in when you plant and watering well for the first few months. Once established it'll give a nice effect with the bonus of spring flowers.
If the main trunks are in the neighbour's garden you are entitled to cut right back to the boundary and you are required to offer the cut branches to the owner. Allowing the branches to remain over 1.5 metres will look distinctly odd and realistically it is a case of cutting back the whole height or nothing. It will look dead and unsightly on your side, though the neighbour's side should continue to grow as normal.
Do try to discuss the matter with your neighbour before going ahead as war could break out. Invite them in for a coffee and to see the problem from your side. Have alternative solutions to offer (eg. Sharing the cost of a new hedge)
These trees are a nightmare. There is legislation in force but it is pretty expensive to lodge a complaint with the council. Best all round to try to come to an amicable agreement. Good luck.
What I was thinking was that a hedge is made up of two densely-leaved sides a few inches thick and the middle, which is largely see-through. If you cut one side off, the effect would be that the hedge would look much more transparent. Is this what you or your neighbour would like?
I have a tall hedge of these conifers at the bottom of the garden because without them we would be overlooked. I didn't plant them but have had no problem getting plants to grow at the base of them. In a perfect world they wouldn't be there but I can't face the wait for a new hedge to grow to a height that would provide privacy. I also grow clematis montana and jasmine through the trees to break up the colour in the summer.
Thank you all for you contributions.
Joe, I see what you mean now, but am wondering how it will look, if I hide the chopped bottom part with a fences or willow sticks On my side. This should not change the neighbour's side.
I also like yviestevie's advice on work what you have. I was looking at the following site which is not dissimilar to what yviestevie suggested with Tropaeolum speciosum interwoven on leylandii. Red contrasts well on green. Beautiful.
one of the pictures on the site I mentioned above.