You might be able to get some help from the high hedges act (restricting boundary hedges to a height of 1.8m) but this will involve surveys etc from your local arboricultural consultant which will again cost money. It is also based on light restrictions to your property, and from the looks of the photo you may not get anywhere. Again, you can only go down this route if your neighbour refuses to act on the trees.
I'm afraid I see this sort of thing all the time (tree surgeon you see) and the most usual / cost effective course for friendly neighbours is a fifty fifty split and remove them.
Please don't try and kill the trees, Leylandii are actually quite hard to kill, and if you do succeed not only could you be responsible for criminal charges, removal of the dead trees will cost a lot more as it takes more time.
By the looks of the photo, you may be able to fell them across you're garden if you take the fence down, so might not cost as much as you may think, but make sure you get a fully insured tree surgeon with NPTC certification.
Hope that helps.
Just noticed the bit about subsidence: what sort of soil base do you have round you're way? I'd only worry if it was a heavy clay, in which case I'd cut them in half, leave it a year or so and then remove the other half.