Lead dpc tray for chimney?

There is damp on the first floor chimney wall of the cottage I am renovating. One builder says the fitting of a proper cowl, removal of plaster and leaving to dry should sort it - especially once the place is lived in again and the stove lit Another builder says it needs a lead tray/flashing and for that the chimney will have to be taken down and rebuild (around £1750).
The chimney and pointing look sound compared to the the neighbours who don't have any damp and don't have a lead tray fitted either! Is this a standard recommendation for chimneys? Can't afford to throw unnecessary money at a problem so torn between getting it done or trying the first option and waiting.

You say the neighbour's chimneys don't have flashing? That seems very odd.
Where a chimney projects through the roof will need some way of weathering the joint- as I understand your situation. And this is either OK or not OK !
The absence of a cowl and rain water going down the chimney is a plausible explanation of damp- as is the failure of the flashing. One thing is for sure- when the cottage was built there would have been a suitable flashing in place so there is no question of fitting a tray as an after market modification.
And as for rebuilding the chimney: it is not necessary to to do this to repair or reinstate a flashing but perhaps the chimney is in any case in poor repair. Very old properties might have lime mortar joints in poor state.
You need the first builder to explain why he thinks the chimney is OK! Is it structurally sound? Is the lead ( I assume lead) in OK condition?
The problem for the builder is the obvious one: faced with a very old property the easiest advice is to rebuild from scratch. Otherwise you patch stuff up and may or may not end up with hassle. Easier to charge a fortune and guarantee no come backs!

You need to diagnose the cause of the damp. A cowl and use will stop damp from the flue, whilst a tray and flashing will stop damp running down the outside of the stack.

succinctly put Woody!

But if a builder is also suggesting a stack rebuild that has to be taken on board. If we are talking over #1500 of work that is a lot stack to be balanced precariously overhead!
I would want to put my mind at ease on that first. You can survive damp.

And down the inside of the stack above the tray

OP, if you posted pics of the first floor damp, and a view of the "c/breast" in any loft space, it would help. Any views of the stack itself & any terminals would also help?

Degraded flaunching is often a cause of damp penetrating down a c/breast.
Poorly installed lead flashing and S&C render (the whole house render appears to be in poor condition) can also be damp culprits.
Do you actually have lead flashing on the stack? The other roof junctions show S&C fillets.

The external pic of your gable shows a distinct shadow of a flue line going up from below the stack - unswept flue soot may have bled thro the render to show this faint flue line?

There's only one pot on the stack but its probable that there are two flues in the deep wall, which, of course, is acting as a chimney breast.

Sorry for the late reply. There is no lead flashing on the stack - to be fitted.

Another late reply sorry. In the middle of a home move and BT has kindly left me with no connection! We took off the plaster on the bedroom wall and the chimney/stove guy came and fitted flashing and soakaways. It seems to have done the trick in that the flue area is no longer getting damp with every rainfall and is slowly drying out. Fingers crossed.