DA - it's best to add the supplementary questions as responses to the original but....
By 'North facing wall' are you referring to the woven fence in front of your neighbours 'hedge'? If that is north facing then you really need to speak to your neighbour about cutting their hedge to a reasonable height as, I would imagine, most of your garden will be in deep shadow for most of the day.
That hedge will be sucking the life out of your garden - as well as the life and soul out of you.
You can cut through the roots that are on your property but you must tell the neighbour beforehand and you must offer them the roots back - they come into the same category as overhanging branches, in that respect.

Don't allow the climbing plants to climb up the conifers though. They would cut off all the light and kill the branches they are clinging to, so you'll have bare branches instead of green ones. And the climbers will want to go though the conifers to the sunny side of the hedge where they will do the same thing on your neighbour's side so you will not be popular.

If you want a screen of climbers it would be best to put up a strongly installed trellis and let them climb up that. The branches behind it would still probably go bare but you wouldn't see them behind the trellis!

Could you say what climbers you have bought please?
Many need sun to flourish. You can add this as a comment under this question, no need to start another one

The variety of conifer will be important too - some stay relatively small and some will get huge - have you checked the potential height of the ones you have?

Thank you Teragram. Great help. Bought Russian Vine and Honeysuckle thinking they'd cover (especially RV which I know grows madly) and I thought it might annoy the neighbours as much as their Leyland annoy me. Haven't spoken for a while to the lady next door, she's lovely and very ill so don't want to upset her, its her awful offspring, thugs who are no stranger to HMP and into drugs. As she doesn't have long to live I'm thinking of holding off on things a bit and then going all out when the house is sold as her late husband took a bit of my land which means I could look from my utility room window into her garden and them into mine. Have dug up that part of the garden and am thinking at the moment of putting pots down after a weed control sheet and pebbles out as far as the roots of the Leylandii go, (have dug them out and left them to dry out. Sorry to go on. Planted in tubs some azaleas and a Rhodedendrum which I believe are ok for a North facing wall.

That sounds like a good idea. I think the Russian vine would have been even worse than the conifers....
Did you use ericaceous compost in the pots for the azaleas and rhododendron?

Thank you for your quick answer Steragram. The azalea and rhodedendrom came in pots and I put some peat free compost with it when I put them in bigger pots, will change it. Yesterday put a string to where the roots have come up from the Leylandii (4ft) and have left the earth uncovered with the roots out which I hope will die. On the other side have put all the pots for now and am thinking of putting down weed cover then pebbles, then the pots for the time being. Bought some Roundup tree stump and root killer which it says allows replanting, I'm thinking of putting that down where the roots of the Leylandii come through to my side. I have a border at the top of the garden where I'm going to throw down lots of wild flower seeds which I love and hope will grow.

The root killer doesn't work like that, have you read the instructions? Its no use just pouring it onto the roots. Normally you have to drill holes in the wood and pour it down them - its intended for killing off the stumps of trees that have been cut down. . You can't just kill all the roots on your side and if you did manage to kill them it might well damage the trees, which sadly are not your property. Even if you successfully get rid of the roots on your side they will grow back again. Were you to remove all the roots on one side of the tree it would become unstable in high winds and might fall, so think carefully before you embark on anything drastic.

This is why i suggested you made a raised bed, which would need to be quite deep.

Re putting down membrane and pebbles "for the time being" - You'll need a several inch thick layer of pebbles to be effective and they will still need maintenance as after a year or two weeds will still manage to seed into them. If there are trees around think how you will sweep fallen leaves off them. And if you change your mind about them they will be a right pain to remove. (How can you tell I'm not a fan of pebbles...)

Sorry to sound so negative but I'm just trying to save you from actions you might regret late - you don't have to take any notice if you don't want to!

Have to agree with Steragram. You are on very dodgy territory by doing anything to this tree without telling the owners. You have proper recourse to law through the High Hedges Act - it may take time but it is the right way to go - not taking matters into your own hands like this, especially when you do not know what you are doing.

I have to question why you are digging down to expose the roots in the first place (regardless of what you are doing once you've exposed them). By the way if you make the trees unstable and one crashes down and causes damage you may well find yourself liable for the cost of the damage - and uninsured!

You need to speak to your neighbours, your local authority and a qualified tree surgeon.

Thank you again Teragram you're being really helpful. Spent most of yesterday looking at plants that will grow near the trees and it seems Bears Breeches which are quite tall, Jack Frost and Mrs. Robbs Bonnet seem ok. Haven't put anything down to try and kill the roots but have left the 4ft of soil bare where the roots go to. The other 5ft. have put lots of largish pots with different plants (some of which I'll plant out when I've sorted everything. My neighbour on the other side remarked how attractive everything looked which was nice. Today I'm finishing of my bit of herb garden and when my daughter gets home will get her to take a photo and put it on here. I really appreciate your help. The raised garden you suggested sounded great and had thought of that but think I may need someone to do that as I'm not as young as I used to be and after looking at different ways of doing one don't think I'm up to it myself. Have left the 4ft bare for now so will ask around to find a gardener and get a price, thats if I don't get the above mentioned plants that like shade.

Dear Teragram. Just found this 'thanks' part again, couldn't see it when I did the edit comment. Takes me a bit of time to get used to things, I put it down to my age (and being a bit thick).

Thank you Urbanite, much appreciated. Have put my latest effort on to Steragram's and, by mistake, the edit bit.