Just been researching this a bit an this is the type of flue that's installed so the image of the underneath of the flue block is most likely what these lot refer to as a 'Cover Block'.
I think that I might tell them to get a chimney sweep in before I start removing the surround as it might just be a blockage (He says hopefully) especially as this was initially about a supposed spillage problem, and I did find part of a wasps nest along with a load of cobwebs, in the ridge terminal.
The metal part of the flue was clear so I didn't think that there might be a blockage further down!
I've taken out the connection between the exit block and the terminal so I might go back and smoke test it without that bit attached to check if there's a blockage somewhere.
Would that explain the heat build up in the surround, just above the fire?
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