Hello BrightBreezyMorning!
The great white throne judgment which will take place at the end of the millennial kingdom, will be the legal process and sentencing of all who have died in their sins. However, scripture states that those who do not have Christ are already condemned as long as they remain in that state:
"Whoever believes in the Son has eternal life. Whoever rejects the Son will not see life. Instead, the wrath of God remains on him.”
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The great white throne is just the official due process for what is already known. Therefore, in regards to punishment, it begins at the time of death for the unbeliever. The event of 'the rich man and Lazarus' demonstrates this in that, both Lazarus and the rich man died, yet after the death of their bodies we find them conscious and aware in their spiritual state down in Hades. There was a great chasm separating Hades with one side being a place of comfort/paradise, which is where Abraham and Lazarus, as well as the rest of the OT saints were. And on the other side was the rich man and the rest of the unrighteous dead in torment in flame. By telling sharing this event, Jesus gave us a glimpse into what happens at the time of death. Today, for those who die in Christ, their spirits depart from the body and go to be in the presence of the Lord (2 Cor.5:6-9, Phil.1:23). For those who are not in Christ, their spirits depart from the body and go down into Hades to begin their punishment in torment in flame. The final judgment will just be due process. God doesn't need to first judge then administer His punishment.