Quite possibly airborne grease, Gran.
Air inside the house has water vapour naturally suspended within it. Especially in bedrooms.
Particles of grease are similarly suspended. As you can imagine, cooking is the main reason for this.
Water vapour and grease will condense on any surface that it colder than it is. Your gutter problem probably induced a "cold chill" effect on the wall.
The temperature of your wall needs to be raised above the point where condensation takes place. Of course, ventilation is essential,
but, there are really only two ways to go with the wall.
You either have to greatly increase the insulation in the wall, or introduce a higher level of heating.
Increasing insulation means that you can probably get way with your existing heating level. No need to add more.
No easy fix though, is there? It's just what we live with in this country with our miserable damp climate. In the old days, draughty old houses with big open fires didn't have this problem. Modern houses with bags of insulation don't have it either.
It's just the ones in between ;o(
Meanwhile ... apply stainblock and re-paint...
https://www.screwfix.com/p/no-nonsense-stain-sealer-white-400ml/31649?kpid=31649&ds_kid=92700047154135765&ds_rl=1243321&ds_rl=1241687&ds_rl=1245250&ds_rl=1245250&gclid=CjwKCAiArJjvBRACEiwA-Wiqq4yAHon3z4lNXCA5ZJdjkLJSQjljVBANvyPhUztNkrhjuaFwL3abtBoCxh8QAvD_BwE&gclsrc=aw.ds