The source of the damp was inadequate flashings on 3 chimney breasts. This has (hopefully) now been fixed, but the chimney breasts are all very damp (40% moisture content according to the moisture meter we bought from Amazon - not sure exactly what that means, but apparently over 20% is bad).
Now running a dehumidifier in one of the rooms, planning to do that for a week, then switch to the second, and then the third. Does anyone know if this is sensible?