I walked past the hut again a few days ago, and had a cursory glance around…I found a piece of guttering in the yard. It had come from the far right of the back of the building (including the top of the down pipe). Seeing as the roof of the toilets is fairly accessible I scrambled up and put it back in place, though i’m not sure it’s all that solid, it should be fine for a while. However, whichst I was up there I noticed that the bottom right few slates on the back section of the roof have come off, leaving some of the timber on the eaves exposed (and already fairly rotten!), and probably allowing water into the corner of the loft.
So, yeah, the slates could probably do with being looked at by somebody who knows more about it that me, the gutter should be fine for a while though…