The process is the same, it's just a different name. Jesus also calls it being born from above. It's the same for both Jews and Gentiles. The only difference is that for the Jewish believer it is a second birth of God, for the Gentile it is the first birth of God. Peter writing to Jews speaks of them being born again, while John writing to Gentiles speaks of their being born of God. Same process, different name.
Paul may have not expressly used the term "born again" or "born from above" but I believe Peter does
1Peter 1:1 Peter, apostle of Jesus Christ,
to the elect sojourners of the Dispersion in Pontus, Galatia, Cappadocia, Asia, and Bithynia,
1:2 according to the foreknowledge of God the Father,
in sanctification of the Spirit, unto obedience and sprinkling of Jesus Christ’s blood: Grace to you and peace be multiplied.
1:3 Blessed the God and Father of our Lord Jesus Christ, who according to his great mercy
begat us again unto a living hope by the resurrection of Jesus Christ from the dead,
1:4 unto an inheritance incorruptible, and undefiled, and unfading, reserved in the heavens for you,
1:5 who by God’s power are guarded through faith unto a salvation ready to be revealed in the last season.
verse 1:3
(DRB) Blessed be the God and Father of our Lord Jesus Christ, who according to his great mercy
hath regenerated us unto a lively hope, by the resurrection of Jesus Christ from the dead:
(ESV) Blessed be the God and Father of our Lord Jesus Christ! According to his great mercy,
he has caused us to be born again to a living hope through the resurrection of Jesus Christ from the dead,
(KJV) Blessed be the God and Father of our Lord Jesus Christ, which according to his abundant mercy
hath begotten us again unto a lively hope by the resurrection of Jesus Christ from the dead,
(LEB) Blessed be the God and Father of our Lord Jesus Christ, who according to his great mercy
has caused us to be born again to a living hope through the resurrection of Jesus Christ from the dead,
(LITV) Blessed be the God and Father of our Lord Jesus Christ, He according to His great mercy
having regenerated us to a living hope through the resurrection of Jesus Christ from the dead,
(LSV) Blessed [is] the God and Father of our Lord Jesus Christ, who, according to the abundance of His kindness
begot us again to a living hope, through the resurrection of Jesus Christ out of the dead,
(Weymouth) Blessed be the God and Father of our Lord Jesus Christ, who in His great mercy
has begotten us anew to an ever-living hope through the resurrection of Jesus Christ from the dead,