When did the last days begin?

The Bible uses the term “last days” in several different ways. For example:

But understand this, that in the last days there will come times of difficulty. For people will be lovers of self, lovers of money, proud, arrogant, abusive, disobedient to their parents, ungrateful, unholy, heartless, unappeasable, slanderous, without self-control, brutal, not loving good, treacherous, reckless, swollen with conceit, lovers of pleasure rather than lovers of God, having the appearance of godliness, but denying its power. Avoid such people. (2 Timothy 3:1-5)

… scoffers will come in the last days with scoffing, following their own sinful desires. (2 Peter 3:3)

Both of these passages refer to the time before the return of Jesus.

But the three other New Testament passages that refer to the last days (Acts 2:17, Hebrews 1:2 and James 5:3) all apply it contemporarily (i.e., to the first century period). For example

In the past God spoke to our forefathers through the prophets at many times and in various ways, but in these last days he has spoken to us by his Son. (Hebrews 1:1-2)

In the Old Testament, the “last days” are mentioned in Isaiah 2:2; Hosea 3:5 and Micah 4:1 (although the ESV uses the phrase “latter days” in these verses). In each of these cases, the reference is to the kingdom, after Jesus returns. For example

In the last days
the mountain of the LORD’s temple will be established
as chief among the mountains;
it will be raised above the hills,
and peoples will stream to it.
(Micah 4:1)

There are several more places referring to the “latter days” in some translations. But often the underlying Hebrew phrase seems to simply mean “in days to come” which is how many modern versions translate it. Examples are in Genesis 49:1; Numbers 24:14; Deuteronomy 4:30; 31:29; Jeremiah 23:20; 30:24; etc.

So there is no defined period called “the last days”. It is a relative term referring to a time at the end of a long period. In particular, it makes no sense to look for a particular event to mark the start of the “last days”.