I think next door may take the 'mines fine so why do I have to pay for it' stance just like ours. We live in an old house which did have a shared water supply.
Every time they used the bath or shower we would completely loose water pressure. So halfway through a shower or filling the bath it would turn to just a dribble
But seeing as they never had any problems they would contribute to the cost of separate lines. Ended up digging up the back garden and driveway (about 30m) and run a 25mm plastic pipe before the water board would come out to connect it up to the mains.
Swings and round abouts though as we also share a sewage main (main runs through the back garden) with them and when their drain blocks we take the f**k it stance. Their problem they sort it

So don't hold your breath for them paying anything