First thing would be to stop more water getting in – has the chimney can got a china-mans cap or elephants foot on it? If so i would do that as a matter of course – without one it’s a large hole pointing skyward. Elephant’s foot £60-80, cheap cap maybe £20. Plus fitting if you can’t get up there yourself – fitting is easy once your’e up though, 10 minute job. I’d probably think about installing an external air brick or vent at the base of the flue.
It’s conceivable you have something bridging the flue. Id drop an endoscope camera down the flue and see whats going on, you can get one to connect to your smartphone pretty cheap. People stuff all sorts of crap in their flues that they thought was a good idea, cement, rubble, masonry – I’ve seen it all.
What looks like a well pointed wall can sometimes be the opposite, my day job is more often than not repointing gables and it is very common to find a void where the existing mortar has just not been pushed back into the joint. Driven rain will find it’s way in through the tiniest of cracks, given a chance. A few mm of mortar skimmed over a joint won’t keep the weather out for long.