updated to iPadOS 16.3 and lost the ability to do iCloud backups. Receive message "unknown error try again later".

Some are having this issue. See the following Apple Support Community discussion which includes an alternate path to back up to iCloud. Also see Demo's advice regarding sending Apple feedback about what appears to be a bug. The iCloud options which were chosen--such as backup, Notes, iCloud Drive*, etc--prior to the update are working if you directly use the relevant app (or the alternate path for backup), possibly for everyone affected.

Can’t get into some iCloud settings, err… - Apple Community

*I was able to access/download files from iCloud Drive at its location in Files app.

I'm glad you've gotten your issue resolved to your satisfaction (of sorts). And @ Babaganoosh:

It's not been announced that 2FA is now a requirement. Until it is, speculation could cause unwanted actions. I understand why you assume that it's now a requirement. However, your action of enabling 2FA could've opened a path in the OS to the things you wished to accomplish––a path otherwise not normally available even in a healthy OS to those who haven't enabled 2FA.

Unless and until Apple declares mandatory 2FA usage, those who are experiencing the issue would be safe in assuming there's a Bug (best to report it).

It's important to note that the apps that were previously enabled for iCloud––like Backup, Notes and iCloud Drive––still work, albeit in an abnormal way. I take that as behavior of a Bug(s).

To back up manually (others are reporting success for auto back ups as well) see:

Can’t get into some iCloud settings, err… - Apple Community

I am having the same issue and some. Are you able to access iCloud Drive files HomeKit?

Is this Apple's way of forcing everyone to turn on 2 Factor Authentication once and for all? I suspect it might be.

Today's update to iPadOS 16.3.1 fixed the issue for an iPad 9th gen without enabling 2FA.

Greatness!

I can confirm that 16.3.1 fixes the problem - on all iOS devices. :)