I think you say the damp is showing on the fireplace wall? Where the chimneybreast would be if it wasn't on the outside? Not on the plain wall which is thinner? This would suggest water is getting into the flues, most likely rain from the top.
Is the damp near the floor or near the ceiling?
the stain looks to me like clean water. An old well used chimney, full of tar and soot, will give yellow or brown stains; but a bedroom fireplace and flue may have been used never or seldom, so will be clean. There will be a separate flue and chimneypot for each fireplace there used to be.
Can you look in the loft. There may be water marks giving clues.
The flues need ventilating anyway, which will dry out modest amounts of damp. They are open at the top so there will be a flow of air and water vapour rising out and escaping through the chimneypots. An airbrick or similar near the skirting where the fireplace used to be will be fine, though you may discover the builders shovelled rubble in, which will harbour damp and needs to be removed.
The chimneypots all seem to have gas-fire terminals, so the chimneys may not be very sooty.
The stain on your render looks to me like a rust-stain, probably due to a fault on cast-iron guttering. But it is not in a position to make the gable wall wet.
Looking at the age of your house I do not believe you have cavity walls, so there will not be wall ties or CWI.
Do not allow anyone who sells silicone injections into your house. Nor "damp-proof" wall coatings. You have a building problem.
I would be thinking about having the chimneystack removed down to roof level and capped, except that you say you still have a downstairs fire. Was the flashing done with lead, or sticky tape?