If you can see the underside of the slates, then there is clearly no sarking felt, (as it's called). With no felt, you certainly don't need further ventilation.
On your ceilings, I guess the signs of condensation are on the slopey bits. The flat part has 400mm ....... that's fine. I'll bet the sloping ceilings have no insulation at all (very common back then.) Of course there's no way of getting any insulation in there. The ideal solution is to plant insulation of the board type (Celotex, Kingspan) onto the sloping parts, then plasterboard and skim. It doesn't sound as though the owners would be willing to pay for this though.
The mortar on the underside of the slates is possibly a previous "bodge" to try to prevent slates slipping. They obviously have "nail sickness" ie the nails have rusted and are no longer holding the slates.
Here, the ideal is to strip the whole roof, lay breathable felt, new battens, and re-nail or renew the slates.
Environmental health would certainly add weight to your position.
Good luck.