At the Judgment of the Nations, do the condemned go directly to the Lake of Fire?

Will those who are judged in Matthew 25:41-46 appear again at the Great White Throne of Revelation 20:11?

Matthew 25:41-46 describes the Judgment of the Nations, which is to be distinguished from the Judgment of the Great White Throne. The Judgment of the Great White Throne takes place in eternity, after the Millennium. The wicked dead will be judged and consigned to the Lake of Fire (Revelation 20:11-15).
The Judgment of the Nations, or Gentiles (the Greek word can mean either), takes place on earth when Christ comes to reign, as verse 31 clearly states: “When the Son of Man comes in His glory, and all the holy angels with Him.” If we are right in identifying it with Joel 3, the location is the Valley of Jehoshaphat, outside Jerusalem (Joel 3:2). The nations will be judged according to their treatment of Christ's Jewish brethren during the Tribulation (Joel 3:1-2, Joel 3:12-14; Matthew 25:31-46).

Does this judgment teach salvation by works? The message of the Bible is that salvation is by faith and not by works (Ephesians 2:8-9). But the Bible is just as quick to say that true faith produces good works. If there are no good works, it is an indication that the person was never saved. So we must understand that the Gentiles are not saved by befriending the Jewish remnant, but that this kindness reflects their love for the Lord.

The Scriptures do not, so far as I am aware, give us any further information about them than what is contained in Matthew 25 itself. In Matthew 25:41 the command is, “Depart ye cursed, into everlasting fire prepared for the devil and his angels”; and then in verse 46 it is stated, “These shall go away into everlasting punishment”; both of which expressions can scarcely be understood otherwise than of the Lake of Fire.

And if they are sent from this judgment to the Lake of Fire directly, it is difficult to see why they should be brought back from it to stand before the Great White Throne; especially as that appears to be entirely a judgment of the dead, whose bodies have been in the earth or sea, and whose souls have been in hell. According to Revelation 19:20, the beast and the false prophet will be cast alive into the Lake of Fire at the Lord’s appearing; and therefore there would be no difficulty in assuming that these of Matthew 25 may be similarly disposed of.