When Christ returns He immediately destroys the anti-christ, this is why He returns at that very time to save Israel from total destruction. Zechariah tells us that 2/3 of Israel will die in this battle with the anti-christ. But there is another threat to the anti-christ from the nations who are trying to help Israel, so he pulls out of Israel to take care of them, thinking he will come back and finish off Israel easily.
When he returns to deliver the final blow to Israel, finishing off the 1/3 left, Israel calls out to God for mercy, and to their total dismay their Saviour is Jesus Christ, the One their fathers crucified. This is at the Second Coming.
The next order of business for Christ is the judgement of the nations, the scripture you quoted in your post, Matt. 25:31-46.
The sheep and the goat nations are separated by Christ. The sheep nations are the nations that aided Israel or at least did not raise their hand against Gods chosen people, the goat nation are the nations that warred against Israel in the this battle, the Battle of Armageddon.
Now the reign of Christ on earth begins with all nations on earth under His rule for 1000 years.
The 34th vs you quoted concerning "them on His right hand" the sheep nation are welcomed into the reign of Christ for 1000 years. In vs. 41 we see the goat nations, those on the left hand, condemned by Christ into everlasting fire.
Most theologians claim that the leaders of these goat nations that warred against Christ will be executed on the spot, but the people of those nations will be granted to live during His reign.