What was the what was the outcome of the third stage in God's plan for the return of the exiled people of Judah?

The stages is what is confusing here. There was the first time the Israelites entered Israel under Joshua. Hundreds of years later, they were temporarily exiled to Assyria and Babylonia, and permitted ot return after the time of Cyrus. The poorest Jews still remained in "Israel" while the majority were exiled. After the Bar Cochba rebellion of 132-135AD, they started to drift away from Israel settling around the world as they were barred from even seeing Jerusalem. The third time they have returned is after the second world war, if that is what you are referring to.

These are not commonly referred to as stages. Nevertheless, for Christians the Jews were expected to return back at some time. Jesus said the Jews in Jerusalem wouldl not see Him again until they acknowledge Him as Lord, "as He who comes in the Name of the Lord" (Matthew 23v39). Therefore for the prophecy to be fulfilled they had to return. "All Israel will be saved" (Romans 11v26). So the Jews will remain in Jerusalem until Christ's return, before He does that many will become Christians or Messainic Jews. Of-course the present Jewish Orthodoxy do not accept this but already many have become Christian in recent years. Figures of membership of Jews for Jesus for example have steadily risen. I hope this does answer your question.