If it's been well aired, freshly painted, well heated and had a dehumidifier running then it possibly won't smell damp. Just wait until you're living in it, with furniture near walls reducing airflow, people in it breathing out water vapour on their breath, washing drying on the airer, cooking, showers and baths, windows minimally open because it's winter and cold. Then wait for a week or two of downpour rain and come back and tell me there's no mould anywhere. Can you tell I've been had before?! 😩
It's 100 year old house so I'd treat it in general however 100 year old houses were meant to be treated. I'm not a builder so I'm going to guess that's something like open working chimneys with a fire regularly lit in them, no cavity wall insulation because the walls were never designed for this, the correct paint and plaster for the materials it's built from so it can properly breathe not whatever is cheapest in the DIY store, windows opened for an hour daily at start and end of day because super-efficient draught-free double glazing hadn't been invented back then so it's meant to have constant low-level airflow not be sealed up 24/7. I'd also assume unless proven otherwise that any path, steps or flowerbed outside that was installed at any time after the house was built, has breeched the damp proof course (because I have a tendency to look down and I've seen so many buildings from just walking down the street where you can see the airbricks to the basement are partially blocked or the damp proof course is either half buried, or only 2 inches below, the new flowerbed or concrete path etc, so I think it's quite common).
I've no idea how much it'd cost that's why I'd stay away from a building with rising damp (or any damp TBH). It's not worth the health risks of living in a building that's mouldy. The mould is there long before you can see it.
There is a reason this property is coming with a quote for a bodge job repair. If you're selling a property you've likely got some equity in it and will get a better price and quicker sale if it doesn't come with a problem, so if the repair was really going to fix the problem then why not just pay the £4k and get it done? Maybe because it's a massive inconvenience to live there whilst it's done or because it won't fix the problems and they don't want to get sued by the new owners? IDK if you could sue over something like this I'm just guessing.
Is it really ground floor flat, as in the level of the pavement on the street outside is no higher than the level of the lowest floor? Because basements were never designed to be lived in, are expensive to properly convert into being properly habitable and there's lots of them around that are bodge jobs. I'd never buy a basement flat full stop.