I would regard the shared chimney as being a party structure. Anything which is a party structure will fall within the procedures set out in the party walls act if the property owners do not come to any decision.
I would not say it is your responsibility to do the works but you can choose to do so. The same applies to your neighbour.
You may consider a 50/50 split on the costs to repair the chimney as being reasonable but rectifying the damage caused by damp in the neighbour's bedroom is not something you wish to contribute to.
If you cannot reach agreement then either of you can rely on the party walls act procedure. This will involve service of notice and potentially a counter notice from the neighbour, if there is a dispute the matter is referred to a surveyor for determination, who will also decide on who is liable for the costs and in what proportions. It is possible to appeal the surveyor's determination by appeal to the county court.
If the neighbour will be on your roof to carry out the repair, if he is to do it himself, then this will be a trespass. He cannot go onto your roof without your express consent even if it is to repair a party structure. That is trespass.
In summary, you and the property owner are free to decide who will be responsible and how costs will be shared. In the absence/failure of agreement either can rely on the party wall act and serve the appropriate notices, a surveyor will determine the matter if there is a dispute. The neighbour will be trespassing if he goes on to your roof without consent, unless he has validly served a party wall act notice on you.
I hope this helps.
Do let me know if you have further questions or queries in relation to my reply. I am happy to assist you further with this.