West walls = subject to UK prevailing SW winds & driving rain. Never cavity fill such walls in exposed areas of country, ie. places with a fair bit of rain... Many older cavity walls are clogged with brickies mortar droppings that sit in clumps in the lower courses & on the wall ties, which bridges the damp from outer to inner walls. Draws damp in & contributes to ties rusting, swelling & cracking the mortar lines. As well as damp inner walls. That said, it could just be condensation from lack of ventilation. Breath / body moisture + bathing, washing, cooking with lids off. All that warm air condenses on cold walls & ceilings. Worse when no vents. You can get passive wall vents that have a batt of rockwool in them so you don't get the draughts; unlike old-fashioned vents.
If cavity walls then even 50mm of internal insulated plasterboard on coldest external walls makes a big improvement to warmth & reduced damp. If solid walls, they must breathe (no gypsum plasters etc!).