How does the Jewish phrase "born again" relate to Yeshua’s new usage?

To answer your question, the term "born again" actually had six meanings in Rabbinic Judaism:

1. Gentile converting to Judaism
2. Being crowned king
3. Bar mitzvah (Jewish Confirmation at 13)
4. Being married
5. Being ordained a rabbi
6. Becoming the head of a rabbinic academy

At what time were the 12 apostles born again? And how could a Gentile be saved before Messiah came?

During Messiah's lifetime, whenever any person actually came to believe that Jesus was the Messianic King, then that person was born again. Before the crucifixion, one did not need to believe that Christ died and rose again to be saved, as it had not yet occurred. But one had to believe that Jesus is the Messianic King and the God-Man. At some point, 11 of the 12 apostles came to believe just that. John came to believe it very early (as early as chapter one of his gospel), and the others a bit later. By the time of Matthew 16, all but one of the 12 disciples had been born again, all having come to believe that Jesus is the Messiah, the Son of God. However, Judas never came to that conclusion and, therefore, was never saved. The way a Gentile could be saved before Yeshua's time was by putting his faith in the God of Israel as being the only God, and rejecting idolatry. It would not be necessary for a Gentile to convert to Judaism to be saved, but simply to put faith in the God of Israel as the only God and Savior while rejecting all idolatry and, then, living under the laws of the Noahic Covenant (Gen. 9:1-17).