Once you've made sure the chimneys are capped with vented cowls (if they're not, a lot of water will be coming in - think how much rain accumulates in a bucket left outside) and then read this from a damp specialist company
https://www.tracebasementsystems.co.uk/post/damp-chimney-breasts
Ventilation can work, but only if no part of the chimney is colder than the dew point of the air you are letting in. You can use a hit/miss vent and take the time and trouble to assess whether there is a condensation risk at that time and open/close the vent accordingly, but on balance because the dew point of room air is about 12 degrees, and the loft/top of chimney is very often colder than this, you have to be very careful with through-flow ventilation because you can introduce more problems than you solve. If it's 15 degrees or warmer outside, yes, ventilate, but if it's not, then don't.
The best place to ventilate a chimney for through-flow is to outside, because the air being drawn in is already at ambient temperature, and as there is no scope for further cooling inside the chimney you won't get condensation.
To those who still persist in suggesting that bottom ventilating a chimney is always a good idea, I ask "why do we fit condensation traps at the bottom of vertical air extractor chimneys?".....https://www.extractorfanworld.co.uk/blogs/condensation-traps