looking at the front view, I wonder if the terrace was originally built like that, or the end was demolished, for example after war damage?
The gable wall looks as if it has had water where the lower roof meets it. Good leadwork will prevent that. Black paint is not so good.
You say the chimneys are capped, so they might not have through ventilation. Hold a joss-stick or cigarette in front of the air vents and see if there is a draught sucking up the chimney (there should be). You can unscrew the grilles and have a sniff for damp if you like. If that turns out to be the trouble, it is possible to ventilate the tops with an airbrick, or even drilling into the flue from the loft, if you are certain the fires will never be used again. If you are considering spending money on the chimneys, ask about having them taken down below the roofline and the slates/roof reinstated above them. I have known this save a lot of future leaks and maintenance on chimneystacks, and the flues can be left open.
Without being there, I'd be inclined to think that it is rain getting in round poor flashing. This should be visible in the loft. If you know a keen photographer with a telephoto lens they might get some closeups of the stack and the gable wall. I can see it will be awkward to get to. I had a roofer round yesterday who tells me he will have to hire a cherrypicker for a day.
If you are really lucky and it turns out to be condensation that will be easier. A bit of damp in the flue will evaporate away with good airflow.
edit
I agree with Ree that lead is what you need, not s&c, if that's what it is. Any builder can do mortar, but good leadwork needs a bit of skill. Lead would last about 30 years, but less than 100.
p.s.
I think the gutter has been leaking or spilling out of the end