Is it an old terrace with a gas fire?
Top tip here; I've had a few over the years that appear to have a damp problem at first glance, and even a few supposed 'experts' have concluded they needed new DPC's. However, on more than one occasion it's been nothing more than condensation, caused by the lack of sufficient ventilation post the blocking off of the old open fireplaces and the addition of efficient double glazing.
it might be worth hiring a dehumidifer for a couple of weeks and seeing what effect it has before ripping the place to bits