If you are using one of the new Macintosh M1 systems, you might want to pat close attention to this. According to an article by by Tim Hardwick and published in the MacRumors web site:
“Apple’s built-in Time Machine backup solution for macOS appears to be causing problems for some Mac users running the latest versions of Monterey and Big Sur, based on a steady trickle of reports on both the MacRumors Forums and Reddit.
“While some users are complaining of different issues with Time Machine on different Macs and versions of macOS that are hard to replicate, one common complaint in particular has surfaced regarding Time Machine backups not completing for M1 Mac users running Monterey 12.0.1 or Big Sur 11.6.1.
“The issue seems to occur when Time Machine runs its first backup after either Monterey/Big Sur is first installed or the operating system is updated to the latest point release. Time Machine says it is “Waiting to Complete First Backup,” but as it appears to be reaching its conclusion, Time Machine suddenly reports “Oldest backup: None” and “Latest backup: None,” and then fails to offer any notice that the initial backup has successfully been performed at all.”
Details may be found at: https://www.macrumors.com/2021/12/08/time-machine-initial-backup-error/.