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It is rank in there. He's quite old and I help him clear stuff out from time to time, so I see lots of piles of stuff. What could I do about that if so and how could I confirm? He's not going to be interested in doing anything himself. Presumably I'd want to block it off at the stone:
Clean back the stone (the entire wall or just the currently affected area?) so there's minimal loose dusty stuff.
Treat the stone (pva?)
While I'm there put in some damp injection cream near the bottom of the wall to be sure.
Wait for a while (how long?)
Fill it then skim or (or plaster board somehow (maybe screw in battens into the stone (though presume screwing would wreck the damp proofing?)
None of the above?
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Profit?
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What could I do about that if so and how could I confirm?
I think you might just have to talk to him and see if it helps by offering your own help.
You've basically described what damp proofing is, and I think most people believe that it's just something pedalled by snake oil sales people. We did have it done, and our damp issues seem to be sorted but we did improve things on many fronts at the same time. I was annoyed with myself for just trusting the survey recommendations without checking if it was real, but it was the blind leading the blind at that point.
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Rising damp is real, but incredibly rare. It is commonly misdiagnosed. You're more likely to have a compounding damp issue (from old mortar, or window detailing), or just plain old condensation. Unless your house is built on very boggy ground or has encroaching water lines?
The party wall situation you're describing sounds like it might have poor breathability and that the wall might have a hard time becoming warm/dry again once cold/wet?
What mortar is present/being used on your neighbour's side? Has the external render changed since being built? Are any plants growing there? Is air free to circulate? Any guttering issues? Has the ground level increased against the wall? Have any air bricks/vents become blocked up? Any other obvious leaks/problems?
It's not terrible (and quite common) for external brickwork to become wet in Winter and then dry out again in Summer, the key thing being that the drying out actually happens, especially when there's no cavity.
The most popular/effective solution is an external application to the bricks so they stay dry (and the wall can still breathe), but I doubt that's possible in your situation and is probably overkill anyway.
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None of the above?
This, mostly.
Strip back to brick and leave for a while.
Then you can see what the problem actually is (without moisture wicking around because of the mix of porous and non-porous, modern and old materials).
Using pva / creams / modern fillers on your side will mask the problem for a while (even years), but ultimately make it worse as it leads to damage of the bricks & mortar.
Otherwise, mostly what @JonoMarshall says.
As far as external applications, I'd want to avoid anything that wasn't lime based.
From this it sounds like the damp might be coming from the neighbours kitchen /utility guttering behind the wall?