As you probably know, unused flues should have some sort of ventilation. It helps in these kinds of circumstances.
A "rain hood" or something similar (clay pot-like thing with a cover to stop rain, and vent holes in the side.)
But, back to the plot... 1930s can be a bit of a transition time. Before that, stacks may have been built with a horizontal "tray" just above roofline. Probably pieces of slate.
This tray is like a DPC in reverse. It stops water going downwards instead of upwards.
It may have no tray at all, or the tray has failed. You usually get away with it, but with recent weather.....................?
Anyway, before taking down the stack and re-building it with a proper, stepped lead tray... Try this................
https://www.toolstation.com/feb-silicon-external-waterproofer/p56806?store=HK&utm_source=googleshopping&utm_medium=feed&utm_campaign=googleshoppingfeed&gclid=EAIaIQobChMIhfWk0eTs5wIVCLDtCh0MDgikEAQYASABEgINk_D_BwE
There are loads of other makes available.
I've used it a few times. A couple of coats all over your stack will do it.
If things improve, then you can think about a stack rebuild.
After all, I may be quite wrong. It may be something else, but this is a relatively cheap way of finding out.
The silicone treatment may well last for several years anyway.