It's actually chapter 24 verses 29 and 30 where Jesus starts to explain when, exactly, this separating work takes place and, clearly, it has not begun yet. There are phenomenal celestial signs, with the sign of the Son of Man appearing in the heaven. The tribes of the earth are smitten as they "see the Son of Man coming upon the clouds of the heaven, with power and great glory." That is when Christ's chosen ones are gathered from all parts of the earth.
More information is detailed in chapter 25. Christ unexpectedly comes in his glory, with all the angels, and Christ sits on a glorious throne. Before him are presented all peoples of all nations, "and he shall separate them from one another, as the shepherd doth separate the sheep from the goats. (vs.32)
This marries in perfectly with Revelation chapter 14. Christ's chosen ones being gathered from the earth to him corresponds with vs. 15 - the first reaping of the end-time harvest, which immediately sees the second reaping of that same harvest, when all others are cast into the wine-press of the wrath of God.
In order for all the living and the dead to be gathered before Christ's judgment throne, they all have to receive resurrection bodies. Those are bodies made fit for whatever eternal destiny they are judged to have coming to them (Revelation 20:11-15).
Here is how an exposition of the book of Matthew puts it, with the heading "The Prophecy of Messiah and the Kingdom, chapter 13" before linking that in with comments on chapter 25:
"Only at the resurrection of the dead with the manifestation of the
wheat and the tares, the sheep and the goats, will it become clear
what was and what was not of the kingdom. So that not until the day of
judgment would all that is of God be gathered to glory out of all the
generations which preceded, from first to last.
...These seven parables foretell the inception, infiltration, and
permeation of corruption, hardly visible in the beginning but
overwhelming at the end. The parables give the history of the sowing
and effect of the word of the kingdom throughout time till the day of
judgment...
The coming in of the multitude of these usurpers - at first
indistinguishable in the common reception of the same seed - would wax
worse and worse, till the kingdom at the end of this present age would
bear no resemblance at all to its appearance in the beginning.
Judgment, terrible and inexorable, fiery and overwhelming, immutable
and everlasting, will appear in the coming day of retribution.
This will result in an eternal separation of the true from the false -
the sheep from the goats - alike raised from the dead out of all
generations, reaching even to that selfsame hour in which Jesus spoke
these prophetic parables as he sat in the ship addressing the
multitudes on the sea-shore in Galilee." The Evangel According to
Matthew - An Exposition, pp 25-26, John Metcalfe
Then comes connecting thoughts about Matthew chapter 25, wherein:
"Jesus reveals the apocalyptic vision of the end of the temple, of
Jerusalem, and of the world... That day of wrath and judgment will be
as sudden and unexpected as was that of the flood in the days of Noah.
The judgment will be as inexorable, falling upon every faithless
servant. Of the ten virgins, the folly of the five will exclude them.
Of all the flock, not one of the goats shall escape that judgment
which is according to works, despite their convictions to the
contrary, and for all their loud and loving profession of Christ.
Every one of these shall go away into everlasting punishment. But the
righteous - for that is the character of the sheep - shall enter
into life eternal." (Ibid. pp 43-44)
The main point for this answer is the totally unexpected timing of the start of a series of events, with no gap inbetween them, culminating in the separating of the sheep from the goats, before Christ's throne of judgment. There will be a tipping-point in world events, just as there was in the days of Noah, when the two reapings of the one end-time harvest shall begin, and all the living and the dead will receive new, resurrection bodies to stand for judgment before Christ. The sheep are separated from the goats at that point.