I see, so the problem is far worse, just doing an paint over it that is damp proof will not solve the problem.

NO, paint will not cure this problem. It is serious and has its roots in the exterior. If you are not in charge of maintaining the building, you need to bring this to the attention of the building manager.

I own this house but I have never seen this problem before since I only got this house few years ago and it is my first house.

You haven't seen the problem before , because the conditions did not exist before.

Leaks develop over time. In this building I would suspect a roof leak. Note that it may not be directly over where you're seeing the problem; water can run a considerable distance before becoming visible.

Given the large wet stripe down the wall I would also check that there isn’t a clog in the gutter drain up top. Causing all that water to flow down the wall when it rains.

@JonCuster I think your comment qualifies as an answer. The only thing I would add is to suggest the OP looks at the gutter during a rain storm.

" You will need to have the wall opened...framing...": given the age and location of the house, the wall is likely to be brick or similar on the inside, brick on the outside, ideally with insulation in-between - assuming it's not just solid brick. Apart from that, what you say holds: seal the outside, wait for the wall to dry out (which could take months), then redecorate.

The wall looks like a solid brick wall to me; chances are there is no (fluffy) insulation, and no framing; these are ideas form Aerican cardboard houses ;-). That house is probably half as old as your nation and would qualify as a major landmark in the West.

This is all speculation. It is unknown until the wall is opened from the inside. We have no idea what has been done in the history of the structure.

@RMDman It IS a solid wall. You can tell because brick ends are visible in the outside photograph.

We still have no idea what is behind the wall sheathing on the inside. Until that is removed all and revealed, everything is speculation. I am not doubting the exterior is solid brick.

@RMDman the spots are actually mould in this case. :-D

As Peter Bill says, that's very probably a solid wall.  Cavity walls weren't common in the UK until the 1920s, so you wouldn't expect this 1900 property to have them, even without the brickwork pattern.

@gidds How would the wiring behind the (what the OP calls) drywall typically be installed?

@JimmyJames It could be clipped to the wall and (wet) plastered over; possibly with capping over the cable (to stop the plasterer’s trowel slicing the cable).

Why is anyone concerned about the brickwork pattern? There is some type of sheathing on the exterior wall. What is behind it is unknown. It may be nothing. It may be some type of insulation and then the sheathing. The point is, We do not know. The wall must be opened to dry out and then assessments can be made.

@John If it is actually drywall as the OP has tagged the question, we would expect some sort of gap between the brick and the drywall, perhaps with some sort of insulation, maybe?

@RMDman With respect, your answer (and repeated comments) may be correct for where you are, but are completely incorrect for typical construction methods in the UK. The brickwork pattern indicates that the wall is 9” (one brick length) thick and therefore very likely to be a solid wall (as would be entirely expected for that period). It’s highly likely to be wet plastered, maybe replaced with dot and dab plasterboard if it has been recently re-plastered. No sheathing, no insulation, nothing to open (short of knocking off the plaster).

@JimmyJames I strongly suspect the wall was wet plastered, with no gap and no insulation. As I mentioned above, there’s a chance the plaster was knocked off and replaced with dot and dab plasterboard. Accordingly, I suspect the drywall tag is incorrect.

@RMDman I think the point is that those 'short' bricks are typically full bricks but placed perpendicular so that they 'bond' to the inner brick wall. So their presence suggests that is the construction.

@ Jimmy James, It is all moot since the wall covering on the inside must be removed. Unless someone comes up with a viable reason that it cannot be removed. I.E. the wall inside is solid concrete bonded to the brick exterior and smoothed to look like drywall.

@RMDman I agree with you more or less. I mean, drying out brick is a little different than drying out e.g. old doug fir. But for sure any medium that is growing mould (mold) and/or mildrew that well needs to be removed with extreme prejudice. Aside from that, I'm just explaining why I think they are getting all butt-hurt about your answer.

@JimmyJames, LOL...it seems a lot of people get "butt-hurt" about a lot of things that don't matter anymore.

@RMDman Should we tell them that a house built in 1900 isn't that rare in the US? It really isn't that long ago.

@JimmyJames, And just because it was built in the 1900 doesn't mean that it has retained all the properties and has never been upgraded or remodeled.

@RMDman: These properties are super common in the UK. Many cities will have row after row of them. I, myself, lived in a very similar one for years. The part shown there is actually an extension, but the bricks clearly indicate its age as similar to the main building. In theory, yes, it could have been upgraded inside but it's hugely unlikely - that kind of modification is very, very rare without rebuilding the entire extension better.

@JimmyJames OP didn't add the "drywall" tag; Machavity did. I've left a comment under the question, querying this, and asking OP to confirm either way.