Right the two internal walls either side of the stairs, have marks about 500mm up.
Not rising damp then - that starts at the bottom of the walls.
What is under the stairs - possible that a load of stuff in there is attracting dampness (or was - probably removed but the damp remains due to no ventilation)
Or it's just that the previous occupants had some furniture against those walls, and they attracted condensation due to them being colder than the other walls (very likely if the space under the stairs is not heated, or the floor under the stairs has holes into the void below)
His damp meter lit up like an Xmas tree on them, but did not on the external walls.?
So by their own dodgy testing, the external walls don't need any treatment.
Woodworm holes appear when the beetles causing the damage leave - so the presence of holes on it's own just means there was an infestation at some time in the past.
If there is powdery dust in/near the holes, the beetles have recently left, there may be other eggs/larvae left in the wood. Treatment would be advisable.
If no dust, the beetles probably left years ago.
The real issue with woodworm is the condition of the timber - what looks good on the surface can be totally destroyed inside, so some further investigation will be required. Damaged sections should be replaced.
For the damp problem (which from your description is actually very minor and insignificant), the easiest thing to do is put the heating on and leave the windows slightly open to get plenty of fresh air through the building. If the damp is still there after a few weeks, or gets worse, then look for other causes.
The beetles causing woodworm are also more likely to be found in damp conditions, so heating and ventilating the property will reduce the possibility of further infestation.