Chimney stacks repointed, rebuilt or taken down, flaunching renewed, pots refitted, cowls and caps fitted, and lead flashings, soakers, valleys and aprons dressed properly. The chimney is the most exposed masonry on the house and the flashing round it is the single most common source of a leak that everybody blames on the roof.

Sound stacks repointed, stacks that have lost their bond taken down to a good course and rebuilt in matching brick.
Cracked flaunching cut out and renewed, pots reseated, and caps or cowls fitted where a flue is redundant or downdraughting.
Aprons, soakers, step flashings and back gutters formed in code lead and dressed into the joint, not smeared with sealant.
Stacks no longer in use can be taken down and the roof made good, or capped and ventilated so they are not a maintenance job forever.
Nine times out of ten it is the flashing where the two meet, not the roof and not the stack. The lead apron, the step flashings up the sides and the back gutter behind the stack all have to be dressed and lapped correctly, and when any of them lifts, splits or was sealed with mastic instead of lead in the first place, water goes straight in and runs down the side of the flue. It is usually a straightforward fix once it has been correctly identified.
Look at how far the problem goes. If the mortar joints are eroded but the bricks are sound and the stack is plumb, it is a repoint. If bricks are spalling and crumbling, the stack is leaning, or the mortar has gone so far that courses are moving, repointing a stack in that condition is money on a stack that will need taking down anyway. On the coast around South Shields and Whitburn, salt and wind take stacks harder than they take the rest of the house, so leaning and spalling are common enough to be worth a proper look.
It is the mortar fillet on top of the stack that holds the pots in and sheds water off the top. It cracks with age and frost, and once it does, water goes straight down inside the stack, into the brickwork and eventually into the bedroom below as a damp patch on the chimney breast. Renewing flaunching is a small job that solves a damp problem people often spend far more money chasing indoors.
Yes. A redundant stack can be taken down to below roof level and the roof made good over it, which removes a maintenance liability permanently. The alternative, if you want to keep the look of it, is to repoint, renew the flaunching and cap the pots with a vented cap so the flue can still breathe. Both are legitimate, and which is right depends on whether you would ever want the fireplace back and how the elevation looks without it.
Because lead moves and lasts. A roof and a stack expand, contract and shift slightly with temperature and settlement, and code lead dressed into a raked joint accommodates that for decades. Mastic and self adhesive tape do not: they go hard, crack at exactly the point where the movement happens and let water in behind, and then the next person has to remove the lot before doing it properly. Where lead is the right answer it gets used.
Yes. Cowls for downdraught, caps for capping off an unused flue while keeping it ventilated, and bird guards where gulls or jackdaws have decided the pot is theirs. On this coast the gulls are persistent and a nested flue is both a blockage and, for a working fire, a genuine safety problem.