The webcam stopped working. It was probably being plugged into a different USB slot, and the driver was re-installing (why does that happen if you use a different slot?), but the re-install was failing: "The class installer has denied the request to install or upgrade this device."
Anyway, to cut a long story short, and for the benefit of future googlers, the problem must have been that some webcam files had been installed under a different username, and were failing to be overwritten. Having logged in with the correct user, the re-install succeeded, and now it's working for the other user too.
In case you're wondering, both users are administrators, so it wasn't that.
Update: I typed too soon - after unplugging and replugging the webcam it stopped working again.
This time I logged back in as the user it works for, plugged back in, then looked at the list of driver files (via Device Manager). I worked my way through the list, finding each file in Explorer and looking at the properties of each file, and three that were executables had some text on the Properties General tab saying that the file had been copied from another computer and were blocked. For each one I clicked the "Unblock" box and then "Apply", then unplugged the webcam, logged out and in to the other account, plugged the webcam in, and hey-presto, this time it worked.
Crossing-fingers that it lasts this time.