Why will neither of my AppleTVs update to tvOS 16.1?

I have 2 ATVs. One is HD, the other 4K. The former is connected via WiFi, the latter via direct Ethernet.

I have repeatedly tried to update them to 16.1. Invariably this fails after preparing to update with the message, "The update was not successful, try again later" and an OK button. (Don't know why trying again later would change anything, as this happens AFTER the update has been downloaded. )

When I click OK, the home screen shows ALL my third party apps updating (or reinstalling--same behavior). Don't know if this is significant. I've never seen this before.

I've never had any update problems before. Anyone else have, and solved, this problem?