How do I keep next-doors wild ivy at bay, rooting in all directions.

Cut it back again and use a vegetation killer on your side

That's a tough problem because ivy is incredibly invasive. You will have to be vigilant about pulling it up or it will quickly take over and because of the waxy coating it develops vegetation killer can be ineffective. If you choose to use vegetation killer it works best on new growth in the spring before the waxy coat has developed

Spray with straight vinegar. You will need to repeat periodically, but vinegar is cheap and environmentally friendly.

Whether you use something like round up or vinegar, it would probably be better to get the poison to the roots if you cut the stem fairly close to the ground and paint the cut to get around the waxy coating it develops. By cutting it down and disposing of the upper part, you are effectively shortening the time and distance the killer you use takes to get to the roots. Vinegar will kill anything it touches, hence the paint it on the raw cut end, it won't touch the grass or any other plants around it.

You need to dig a barrier into the ground deeper than the roots of the ivy. Say, a 6 or 8" deep strip of corrugated metal. Then kill whatever grows back after digging out the roots of every ivy in your yard. OR you can make the ivy a part of a garden area that you contain with barriers and decorate with rocks or a wagon wheel or a wishing well or a bird bath area. (If you can't beat 'em, join 'em)

Try and speak to your neighbours about the problems you are having,