Anyway, just as pseudomonas was leaving this evening, having dropped round, I heard water falling, asked it was raining and was told no, it was an overflow pipe overflowing. He went on his way with a cheery "you know what to do". It turned out my toilet cistern was overflowing. Further it turned out that there was water spraying out of the joints of the filling mechanism.
There followed several minutes of damp and pointless attempts to tighten it, several more minutes of trying to find the stopcock, a couple of minutes of the stopcock failing to budge, and then suddenly a lack of water. At that point I realised I was rather thirsty so went and drank the luke-warm water that was left in the kettle (see "not very practical" above).
After that, dismantling the cistern mechanism was a much less damp experience and it became obvious that a washer had perished and was no longer toroidal, which was detrimental to containing the water flowing through it. I lack any washers, and know of nowhere convenient to get one at half past 11 in the evening. However, inspiration struck and I realised I do own a small pile of old bike tyres¹. Somewhere. After some searching I found the pile and pulled out one that had entirely blown, was impractical to patch, so should have probably been thrown away at the time. A brief application of scissors yielded what has proved to be a workable rubber washer (or rather pair thereof) that have managed, on the first attempt, to allow water to flow through the right bits of the cistern and prevent it from exiting elsewhere.
¹Edit: Or rather inner tubes, but you probably guessed that's what I meant. I do also have several old bike tyres though.