The dampness is always in the first 18" above the skirting board. The
wall does not have a cavity and it's always cold. I suspect the dampness
in condensation rather than rising/penetrating,

Why do you think this?

It could very well be a failed/missing/bridged damp proof course. This is
easily remedied with a chemical injection. The expensive part is replacing
all the plaster.

If it really is condensation, this can be fixed by additional insulation and
ensuring adequete ventilation.

Christian.