Because I first assumed they would be responsible (again), I sent a screen capture video to our IT department where you can see it take 49 seconds between a right click on a file and the moment the context menu pops up. That was a file on a network share where no svn repositories are located anywhere nearby.
(the screen video shows nothing really relevant to Tortoise, so I'm not including it here.)
Once the menu appears, explorer behaves approximately normal until you close its window. Open a new one, and the problem comes back.
The effect is less severe on local disks, but the symptoms exist there too.
As the week went by, I managed to gradually pinpoint the problem to "something to do with icons", discovered that resetting the icon cache made no difference, and in the end I discovered just now that the problem completely goes away when I uninstall TortoiseSVN, and comes back after the first reboot when I reinstall it.
Now I didn't update or upgrade Tortoise recently, so I think some Windows 10 update that came in over the weekend or early on Monday, or something like that, must also be involved.