This text indicates that God has done something that causes every single person to bow and confess that Jesus is Lord.
Well, that would be the case after the second coming. If you're translating "would" to imply "in the future."
That is a possibility with the subjunctive aorist in Greek.
A Greek Grammar of the New Testament and Other Early Christian Literature by F. Blass and A. Debunner, translated and revised by Robert Funk, states: "For this mixture in late Greek, growing out of the futuristic subj. and favored by the phonetic leveling of -σει with -σῃ, -σεις with -σῃς, -σομεν with -σωμεν, etc., which led e.g., to a purely futuristic use of the aor. subj." (p. 183)
Thus, the subjectives can be understood as wishful thinking about the future, e.g., "might bow" and "might confess" or "should bow" and "should confess". Or, they can be equivalent to the future tense: "will bow" and "will confess". This is what is expected in the future. There are numerous variant readings of Phl 2:11 that use the future tense: "will confess". It's possible that a copyist changed it to subjunctive to match the subjunctive: "would bow".
On the other hand, Isaiah 45:23 (LXX) uses the same verbs in the future tense: "because to me every knee shall bow and every tongue shall confess God." A copyist may have been influenced by this and changed it to match the OT. (But why not both verbs?)