Small rain leaks around chimney stack

Hi, I would really appreciate advice as I'm no roofer. I have an end terrace house with, as the thread title suggests. A roofer came to look late in the day and condemned my roof basically. "It's tired, it needs replaced - £5,500 will do it". It was just a quick peek from ladders and, competent he certainly looks and I believe I would get a good roof for the money. It's also quite possible a busy roofer at this time of year would only be interested in taking on a job if it's good earner. At that price I need to get a second opinion at least. To my untrained eyes it just doesn't look shot and these are only small leaks when it's raining quite heavy.

I'd have said it needs flashing, or re-flashing. Needs new guttering too. These are old terrace houses and downpipes seem to have become a thing of the past. Plastic guttering butted onto wooden guttering clearly not sealing. Everyone else's gutters blocked.. That's a different question though. I know what I want there. Isolated gutter system from the neighbours with a new downpipe installed, into the toilet wastepipe if that's permitted (sure hope so).

I've enclosed an initial photo of the roof I took today. Next door's stack is the one on the left of course. The leak is from around my stack on the right. No roofer as I said but, I can't see much flashing, just what I think are called soakers. Anyway, useful input most welcome. Flashing, or does the roof look knackered? The flip side of the roof doesn't seem to be leaking.

Roof doesn't look to bad from that photo, one of the soakers looks to be slipping out and it looks like you have a slate missing next to the skylight.

The roof plane and the slates look fine - some slates near the stack and the roof light are slightly disturbed.
And there's a possible humping just below the neighbour's stack?

The flashing on your stack definitely needs attention - note, back gutters are often leak culprits.
You have stepped cover flashing and hidden lead soakers.
You have sand and cement fillets instead of lead flashing at the parapet.
The stack capping of flaunching and pots will need examining.

Do you have an occupied loft room? Where do damp signs show?

The roof light being tight to the ridge is not common.

Metal guttering needs to be taken down every so many years and cleaned & painted - the fascia can also then be seen to.
Your gutter could be isolated at the repair.
Discharging into the soil system is sometimes done but isn't your S&VP at the rear elevation?

Late now and I'm up early but thanks for the really useful input. Loft is a double bedroom. It's actually tenanted but yes, they reported it dripping from just down from the chimney stack when it was raining heavy and there's a discoloured patch by the supporting beam in the attic just before the eaves. The guttering is that hollow plastic stuff. Thanks for the terminology too. Helps me identify what I need. I won't be trying to remedy it myself, I'll get a pro in but I never felt like I should be writing this roof off just for a leak around the stack. There's discolouring on the opposite side too in around the same place. Next door's chimney I would guess. Noted where you mention humping, and yes I think I see what you mean. Soil pipe disappears into the ground round the back but the waste pipe comes out of the gable end. Bathroom upstairs - photo

The S&C fillets, i should have mentioned, should be replaced with lead flashing.
I see what you mean with the S&VP gutter connections.
Is there a WC in the attic room?

No, the WC is on 1st floor coupled up just near that window behind the pipe. Wastepipe looks to come out a long way below it from that pic
So the S&C fillets are those slabs above and below the stack making up the edge. Were they an original feature when the house was built? Do you mean flash over them or literally get rid of them?

Ref WC's & soil pipe I was just fishing to find any soil pipe vent piercing the roof out of sight of your pics.

The edging at your gable verge is called a parapet, and where the slate meets the parapet it should be flashed with lead.
Usually, the S&C fillets should be removed, & replaced with lead cover flashing and soakers - the problem is your parapets are so low that there's hardly any height for fixing cover flashings.
It maybe best to leave the fillets alone unless you can definitely see leaks coming through them.

I see what you mean now. I can't say for 100% but I'm sure it's the chimney. I'm due to visit and take another look. Friday would be good because it looks like crap weather.