This is an interesting question, and actually quite difficult to answer.

People in Western Europe would have been aware what time of year they were in at all times. Religious ceremonies and feast days were a constant reminder of the month and day it was. However, there is no reason when celebrating these events to remember what year it was.

For official documents, certainly in the High Middle Ages and Later Middle Ages, the most common for me of dating was through regnal years. Official documents would follow a system where a date would be described as something like “the Tuesday before the feast of St George in the 3rd year of the rule of King Henry V” so even in official documentation the actual year was not directly referenced though it would be possible to work out the year, but ultimately there was no real need to this.

On the other hand, there was a surge in apocalyptic writings in the build up to the year 1000, much like the Y2K phenomenon of our modern era. If the actual date of the year wasn’t important then this would not have occurred.

I think ultimately we have to conclude that it’s possible people knew what year it was, the educated such as the clergy certainly would have. Nevertheless, as the average person goes, they had the potential to know, but probably didn’t really care as alternate systems of dating, like regnal years, had more relevance to their lives.

Sources:

C.R. Cheney, A Handbook of Dates for Students of British History, Cambridge University Press, 2004.

Richard Landes, The Fear of an Apocalyptic Year 1000: Augustinian Historiography, Medieval and Modern, 2000.