There was no switch at the time. The year we designate "year 1" was the 28th year of the reign of Emperor Augustus, traditionally 754 AUC (the dating system counting from the foundation of Rome) or the 3rd year of the 194th Olympiad (on the dating system beginning with the supposed date of the first Olympic Games in 776 BCE). There would have been many other ways to designate this year, as individual communities often dated them by the tenure of particular priests, priestesses or magistrates. It would not have occurred to anyone that a new count ought to begin that year. Since people couldn't have known in advance that Jesus would be born or that his birth would be significant, it would be impossible for them to have a system of years counting down to one. How would they have known what they were counting down towards?

The system of counting years from the birth of Jesus, which has more recently been re-labeled as the Common Era (CE), was adopted centuries after the fact and applied retroactively. You can read more about this in /u/sunagainstgold's older answer here.