At what point do these people gain their righteousness?
At the point they realize that Jesus is their righteousness. He always was.
Every knee will bow and every tongue acknowledge that Jesus Christ is Lord.
If it has knees, it will bow; if it has a tongue, it will confess.
In heaven and on earth and under the earth. (the realm of the dead)
Philippians 2:10-11
that at the name of Jesus every knee should bow,
in heaven and on earth and under the earth,
11 and every tongue acknowledge that Jesus Christ is Lord,
to the glory of God the Father.
1 Corinthians 12:3
Therefore I want you to know that no one who is speaking by the Spirit of God says,
“Jesus be cursed,” and no one can say, “Jesus is Lord,” except by the Holy Spirit.
Romans 14:9
For this very reason, Christ died and returned to life so that
he might be the Lord of both the dead and the living.
Romans 10:9
If you declare with your mouth, “Jesus is Lord,” and believe in your heart
that God raised him from the dead, you will be saved.
Note on "acknowledge" in Philippians 2:11 from Strong's Concordance
S1843 eksomologéō (from 1537 /ek, "wholly out from," intensifying 3670 /homologéō, "say the same thing about") – properly, fully agree and to acknowledge that agreement openly (whole-heartedly); hence, to confess ("openly declare"), without reservation (no holding back).