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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.
NotThamesWater-
When you say external applications - just clarifying it's an internal wall on both sides. But this may be me being thick and not understanding what you mean.
I am wondering if it's condensation related. The room it is near suffers badly from it and I wonder if there just hasn't been enough ventilation in the past. We've got someone at last installing a proper extraction thing there next month. I'll continue scraping away today (though am unsure whether to take back the entire wall or just the affected area. The section currently painted on the left for example feels totally different to the touch/sound (it seems fine).
CYOA
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?
????
Profit?