Victorian solid wall construction - probably 9" brickwork. It does rather sound like rainwater penetration, horizontally through the wall. 9" of masonry, even if it's rendered, can become completely waterlogged........... especially if facing south-west.
I'm assuming that rainwater soaks the brickwork, then shows up in the room below.
The big problem here, is coming up with theories, making expensive alterations/improvements, then finding it was something else entirely.
I think I would try a little cheap experiment. Give the wall above the flat roof a couple of coats of water repellent............
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It's formulated to soak into the masonry, and does work extremely well. It's not a final solution, but remains effective for a few years at least. It's clear, and easy to apply.
Two or three cans of that isn't going to break the bank, and it may well tell us what we need to know.