Damp Issue

I've used a ton of mushroom fixings into cavity blocks in my house, and I've no issues like that, so I can't see it being cold-bridging.

I find it extremely difficult to believe that a 1mm thick plastic cap on the mushroom fixings can magically solve cold-bridging issues....or that cold-bridging is only an issue when it's raining & gets worse the more it rains...or that the standard fixing for rigid insulation is such an incredibly good conductor that it creates such a cold disk on the plasterboard that the plasterboard becomes visibly damp (but only when it rains)....and only in one place in a largish extension...or that cold bridging focused on the small metal disc at the end of a mushroom fixing would cause a damp patch 16cm wide....

So nuts to those people who are claiming all of your problems are being caused by the absence of a 1mm plastic cover at the end two of your mushroom fixings.

I would guess that there's water ingress into the part of the wall into which those particular mushroom fixings were hammered. There's near zero overhang on that outside wall, so not so much protection from driving rain which will penetrate the render after a while. On top of that, the visible crazing in the render is a clear indication that it wasn't done right (something to to do with the ratios of cement-sand-water, or allowed to dry out too quickly or something).

A mushroom fixing is basically a tube with a slot running along it. If it's horizontal, then a small amount of free water entering it at the end in the block could travel back as far as the plasterboard causing the visible issue you can see, expanding more or less uniformly from where it hits.

In the photos, it looks like one damp patch is bigger than the other? I'm guessing that's the bottom one? Because it seems likely that water penetration from driving rain would be worse lower down because gravity.

Solution is the same: better render (even decent paint would probably improve things) to help keep driving rain out of the blocks when it hits the render, combined with something overhead to keep rain off the render in the first place. The something would ideally be a decent overhang which you will see on virtually every exterior wall for a reason, but I'm guessing that's not gonna happen...

Also good luck convincing the contractor who said "Roof overhang? Exterior render? Bollocks to that..." back in to fix the issue his crappy workmanship has caused in the first place.

(PS, "reducing moisture load where possible, improving heat retention & maintaining a good internal temperature and above all else having appropriate ventilation"— are all excellent suggestions all of the time; I just don't believe they'll do anything to address this particular issue).

I'm not about to start recommending anyone starts drilling random holes in their outside walls.

It's difficult to get a sense from those photos of how the roof is put together, but to help keep water out everything should overlap— to keep your feet dry in the rain you should have an umbrella over a raincoat over leggings over wellies. If you skip the umbrella and tuck your raincoat into your leggings and your leggings into your wellies, there's a fair chance the water will run into your crotch right down your legs and your toes will soon be wet.

Same principle with a house. It doesn't look (from the photos at least) that that principle has been applied with that roof. And again, the crazing of the render tells its own story— and it's not the story of a contractor who pays close attention to all of the important details. Same story as told by the lack of overhang— overhangs aren't decorative, they serve a purpose.