A classic way to identify
rising damp is that it only rises 4 foot above the surrounding ground.
The way it works is that water rises through the very small air bubbles entrained in the mortar when the mortar is mixed, but it is fighting gravity.
Chimneys are very hard to waterproof against rising damp as builders tend to throw all sorts of rubbish inside and this bridges the damp proof course.
However, it may be that you have a mixture of wet chimney caused by rain entering the top and exposed sides of the chimney and making its way down to damp proof level, where it is trapped, like water in a sponge, and because the chimney is wet, it is also cold, and this attracts water vapour from the air, that condenses into the chimney.
Start with the easy option, fit a cowl to the top of the chimney to stop the rain entering and paint the exposed parts of the chimney with a silicone based masonry paint, that will stop the rain entering the bricks and mortar.
Keep the temperature of the rooms on both sides steady 24/7. Water vapour is always attracted to a cold surface, you see this on your windows during the winter as condensation, you don't see the same
condensation on the walls because the water vapour sinks in moving towards the cold centre of the chimney.
Keep the rain out and over the warm days of summer with the rooms on both sides also keeping warm, the chimney should at least dry a little, if it is rising damp, then after each lot of rain, the problem will return. If it is rising damp the only solution is to open the chimney and remove the rubble inside and or install a chemical damp proof course. This will not be easy as it requires careful work.