Note: my answer demonstrates an advanced beginner level of knowledge of building science. I know just enough to think I know far more than I most likely really do, so take this answer with a grain of salt and consult a professional as well!
Rand and Shirlock are probably onto something. Let me offer an alternative or perhaps more general hypothesis. Your external walls are made out of porous masonry--brick and stucco (our word for "rendering" here in the 'states). These materials are "reservoir walls" capable of holding a lot of water when they get wet. Since you're in the UK, it's a wet, rainy climate, so the exterior walls can be expected to get wet a lot. Stop the presses!
Such walls have historically worked well because water that enters the wall is able to dry to both the outside and the inside. Outside, sunlight will warm up the wall and dry off the surface, while simultaneously driving water toward the inside in a phenomenon known as "inward solar vapor drive." And heat generated from interior heating appliances dried off the inside of the wall. So where did the water go? It was mostly vaporized and became water vapor.
Now, in old houses, the entire building envelope was really leaky. So excess water in the air could easily leave the house in windy, drafty weather through the myriad cracks and holes in the building envelope. In modern times, we now know that sealing up air leaks improves a building's performance. This may have been done to your house, which I suspect since you mentioned that the historical wood windows have been replaced with modern vinyl ones. However, if we do this to a house with masonry walls and no insulation (you said "The external walls are often quite cold to the touch"), suddenly the humidity in the interior air has nowhere to go. What happens to warm, humid interior air? It condenses into water on the first cold surface it hits. And if your exterior walls are un-insulated and cold to the touch from the interior, that's where the water is going to appear.
What can you do about this? Well, you have a few options:
Insulate the walls. If you add some thermal insulation on the exterior of your walls, not only will they be warmer on the inside (which will make the house more comfortable and also lower your heating bills), but the insulation will go a long way to keep the masonry in the wall from getting wet in the first place. This is probably the best solution, but also the most complicated and expensive. Adding exterior insulation is also very popular and common in Europe, so you will probably find many contractors able to do the job properly.
Keep the walls from getting as wet. Gee, what a surprise. In the UK, wet walls are probably inevitable. But the drier you can keep them, the better. One way would be to add a new exterior siding which includes a "ventilated rainscreen gap." When water strikes the new cladding and inevitably gets driven behind it due to wind and sunlight, instead of draining right into the existing wall, the ventilated gap will cause it to drain down a passageway and exit the bottom of the wall, away from the house.
Get some more interior ventilation. Your damp problem is probably compounded by a lack of adequate air leakage if the building has been retrofitted for increased energy efficiency through air sealing over its lifespan. Air leakage is bad because it robs the house of heat, but it also carries out moisture. There are now mechanical devices called Heat Recovery Ventilators (HRVs) whose job is to provide ventilation while retaining the heat in the air. Such a device, if carefully installed, might be worthwhile.
Remove any existing moisture barriers in the wall. I can't quite tell from the third picture, but that looks like wallpaper. If it's vinyl wallpaper, you need to remove all of it, because it's serving as a moisture barrier. When water in the wall tries to dry toward the inside of a wall with vinyl wallpaper, it gets stuck when it hits the vapor-impermeable vinyl and condenses there, eventually dripping out the bottom.
There are some things you don't want to do that might seem like a good idea:
Don't add any moisture barriers on either side of the wall. If you block off either side of the wall from being able to absorb or release water, you've (theoretically) stopped water from entering, but also exiting! So where's it going to go? Either out the other side of the wall, or it will condense just under the vapor barrier and cause mold, rot, etc.
Don't get a dehumidifier and call it a day. This would be putting an expensive band-aid on a real problem. The machine will be running 24/7 and you'll be constantly needing to empty it. Eventually it will silently break or you'll turn it off for a few days when you're out of the house and the problem will return.
