I finally got around to installing macOS Sequoia 15.5 on my MBA M3 running 15.1. A bit later, I noticed that Time Machine (backing up to a Time Capsule) was failing.
It appeared that the last backup was the last regular backup before I upgraded, and all backup attempts failed after upgrading until I stumbled on the solution.
If I had read the error message in the posted image, I would have saved myself five minutes, because I would have unplugged the Time Capsule before trying anything else. As it was, I wasted five minutes, but “rebooting the server” did the trick.
I’m posting in case anyone else has Time Machine problems after updating, and to ask if anyone else had Time Machine problems after updating. Working from a state of complete ignorance, I speculate that Time Machine was backing up when the updater interrupted it ungracefully.