So, after we'd gotten them settled with beverages, etc. I attacked the tap. After much dismantling, wiping, and looking baffled, I still couldn't figure out whether it was that the inner workings of the cylinder valve were jammed, whether I'd misunderstood how taps worked, or whether taps were actually not mechanical devices after all, but purely magical in their operation, and all this talk of washers was just to hide the truth. I did find the remains of a tiny E-clip, which brought back memories of hunting round town and finally tracking some down in a motor factors' (really). Dug out the spares, and fiddled with those. No good.
Anyway, I decided to have a butchers' at the hot tap, 'cos that worked. First thing I noticed was that the E-clip was differently positioned. Hmm. Copied that, reassembled the cold tap, and it worked! Well, worked better, anyway - we got enough flow to fill a kettle in under five minutes, which was our primary concern.
So now I need to sort out some replacements, before all this happens again. The kitchen's got a mixer tap, and trying to track down a supplier on a bus-route is tricky. Plus, I'm sure last time this happened, I looked into getting a replacement cylinder valve, and decided that Homebase, Do-It-All, and the like now only stocked complete tap units, not any actual components. The Plumb Centre would be a candidate, except that:
 Time to 'fess up: I've no idea what this middle bit's called. Cylinder valve sounds plausible, but I've made it up.
and so when I go to the Plumb Centre's keyword-search catalogue, I feel like Bernard Black.
"Hello? Yes, hello. Right. Is this the place for when your tap's broken and you need a new thing what is the thing that's in the middle of the tap to make it stop and go, only not with all the fancy bits around it and without all the manky bits either, wotsitcalled, the, the, the screwy thing, yes right? Hello? HELLO? Oh GOD!"
Only that doesn't work on the search engine.
And I suspect that our mixer tap pipes are going to be a slightly different distance apart from whatever is the standard now, which will mean spending sixty-to-ninety quid on a complete new unit that then can't be fitted without an actual, you know, plumber.
In other news, the boiler service was finished off a while ago, with a new expansion vessel fitted. I'd no idea how much either of these latter two items would cost, but they came out to about half my finger-in-the-air estimate, so that's nice. Only lots, instead of 'kinloads.