It wasn't the washing machine per se that was the problem, but the stand pipe into which the washing machine dumps its waste:
The water was bubbling out of a join between the U-bend and the vertical pipe, as indicated. This is just a push-fit join, so it's not like I could tighten it. Also, katlinel said it was bursting out of the top of the vertical pipe too. Eak.
I had a theory: the join was lower than the top of the exit pipe. Perhaps, when the pipe was at capacity, this was raising the level past the join? Maybe I could swap that L-bend pipe around, so that the longer part was horizontal, raising the join? Nope: that L-bend is also a converter between the dimensions of the U-bend and the rest of the pipe.
Speaking of the rest of the pipe, I had an alternative theory:
That long pipe in the middle looks horizontal, but the camera's tilted. Looking at the guide, you can see it droops to the right, being unsupported as it is (and it should slope the other way). Maybe there's some sludge there?
Anyway, over the course of an hour, I dismantled all that pipework bit by bit, finding some long-established gunge, but nothing that really explained why suddenly, we had backflow. Such a pleasant hour... Finally, it came down to the last L-bend, attaching the sink's outlet to the waste pipe that disappears into the wall and into the soil-stack, two stories up. And taking that off produced theory number three.
There was one hell of a pile of sludge there, such that it stuck out of the pipe in a wedge. Pulling it out was revolting (about two and a half feet of it), but meant that I could shine a torch down there and see nothing else that looks obstructive.
Basically, it was a mass of fibres and sludge that was wrapped around a small, stick-like thing that could have been a bone, a nail, a bit of plastic, a match... whatever it was, it was wedged across the mouth of that pipe, collecting everything that went past.
So, that was probably the culprit. I reassembled everything, and quickly went to bed (it was late). This evening, after a trip to the laundrette (things have been piling up), I've tried shoving water down both paths, and all seems well. Tomorrow, I'm going to try the washing machine. And I'm going to watch it like a hawk.
But I'm hopeful.
 No, not bunnies.
 Nope, still not bunnies.
 Actually, this time, quite possibly. Forensics would be required, though.