Sorry, just to add that apparently the DPC is two inches above the bottom of the render. Miscommunication between myself and my wife (who spoke to the builder about it).
that's the easy bit, then.
the render will tend to bridge the damp course and allow water to rise into the wall above. But that can be fixed by cutting it back and chipping off the render on and below the DPC.
A builder/bricklayer who does repointing will probably have a small diamond disk cutter suitable for slitting the render so it can be knocked off without cracking the rest of it. There may be other ways.
If you look down your inspection cover (i think it might be a rodding eye), what do you see? It suggests that a drain runs in the ground beside the house, as expected, and this might be the source of the water, for example if it is blocked, and there is a crack causing the water to rise up at the wet point. The render is in contact, or very close to, the ground/concrete. It's possible that rain is flooding that gap and wetting the wall. Have a look with a torch next time it rains heavily.
I'm thinking it may be necessary to excavate (at least) a channel beside the wall; both to see if there is a problem with water coming out of a drain, and, if not, to isolate the bottom of the render from the wet ground. It will be at least six inches wide and deep, preferably a foot. These channels are filled with cobbles or large clean stones, which do not cause capillarity (rising damp) because the gaps between stones are too big, and are usually sloped to let water run away. It is called a French Drain. When you dig a hole next to a broken drain, the hole fills with water if the drain is broken, this is quite distinctive (and surprising if your neighbour has a bath while you're working).
Looking at your pics, I think I was wrong to suspect the roof, it doesn't look like water will be coming off there.