The most probable cause of damp coming through a roof near the chimney is defective flashings at the junction with the roof.
Mortar fillets are sometimes applied as a cheap short-life alternative to lead, but are brittle and prone to cracking and are best replaced with new leadwork.
Sometimes rain can Ingress into an otherwise perfectly good flashing because in older houses there is no damp-proof course (DPC) in stacks.
This can allow water to soak into porous stack masonry and leach downwards into the loft or rooms below.
Damp penetration may also be due to eroded mortar joints or damaged brickwork.