“AUC” was apparently used mostly for historical purposes by latter historians in the Roman world, who cared about the long run.
As pointed out, for “regular” everyday timekeeping by the man-on-the-street (leaving aside astronomical reckonings for the purposes of fixing the dates of ceremony, which would be done by the priesthood), the standard for centuries upon centuries all over the world was counting civil years by the time in office of rulers, 'cause that’s what cut close to home. To this day, in Japan, though the Gregorian Calendar is standard, the AD(CE) year count coexists with the dynastic year count (currently it’s 10 or 11 of Heisei, I forget which)
I s’pose that in Judaea, the locals used a version of the Jewish religious calendar, counting “civil” years by the terms of the High Priest (and ritually it would’ve been in the 3700’s). In the hellenized world of the Eastern Mediterranean there was also a reckoning by a calendar based on Alexander’s lifetime and all these probably coexisted with Egyptian, Babylonian and Persian calendars, but that was essentially a scholarly pursuit.