I have recently acquired a mid-terraced house with a brick chimney.

My next-door neighbour has advised me that he has damp/water running down his upstairs wall, and he claims that a builder/roofer has advised him that my chimney is leaking.

However, I have no water/damp on the walls or ceilings in my home, and I haven't noticed any water falling into my fireplace.

Is it possible that the water in my neighbours home may be caused by a leak in his own roof which is then running under the shingle to mny chimney penetration?

You would be amazed at how far water can travel to find it's way into a structure. It could be originating almost anywhere on the roof and finding it's way to the neighbors walls... Best to call another qualified roofing pro to find it and give their opinion as to weather or not it's your chimney at fault...

It sounds like you need a new flu liner. Over the years, the flu liner in your chimney has deteriorated to the point where the mortar and bricks are exposed to rain water. Over time the mortar joints have become compromised and are allowing this moisture to enter your neighbor's wall (which borders your chimney) from the inside of your chimney through the compromised mortar joints into his wall.
Have your furnace service company check the chimney clean-out in your basement, they will probably find the miniature shale-like reamins of your terra cotta lining.
If it's a wood fire place, have your chimney sweep (who you should be using annually) check out the condition of the interior of you chimney for you.
You will probably be able to have a stainless steel liner installed if your appliances burn gas. For a wood fireplace, I believe your local code may require terra cotta. Also be sure to install a suitable chimney cap which will also keep small animals and birds out during the off season.

Check the cap (the top of the chimny) to see if you have any cracks in it. Check around all the morter joints for craks. Check the flaching and where the flashing is tucked into the brick. These are just afew places to check. If the cap is bad you can install a cover (rain cap). If you can't find any cracks you could cover the top of the chimny for a short while to see if the neighbor still gets water. If not then its not coming from the inside of your chimney. Need to look els where.
Good luck leaks are hard to find.

My neighbours chimney breast is leaking into my home and has caused damaged to my hallway ceiling and wall. The wallpaper will have to be removed and the ceiling replastered. I think the insulation and floor board above have been affected too. I have told my neighbour about the problem verbally and have written him a letter advising him of the water damage. He has failed to act upon my request to fix the chimney and cover 100% of the costs of repairing the damage to my home. My insurance company say it is a legal dispute and I will have to sort it out myself even though I have Home and Content insurance. What do you suggest?