Is the tripartite God the judge in the Old Testament, and Jesus the appointed only judge in the New Testament?

Your question necessarily depends upon the Father being separate from the Son, so any group of Christians who subscribe to the Nicene Creed, which would include Catholics, Orthodox, Anglicans, Lutherans, Methodists, and perhaps Reformed communities reject this notion as heresy, believing that as the Creed says [emphasis added]:

"I believe in ONE GOD, the Father almighty, maker of heaven and earth..."

and

...being of ONE SUBSTANCE with the Father, God of God, light of light, very God of Very God...".

Your first statement makes an incorrect conclusion; there are many other instances where God's judgment is delayed in the Old Testament: Genesis 15:16 ("...the sin of the Amorites has not yet reached its full measure") & 2 Kings 20:16-19 (Judah's captivity by Babylon is prophesied to occur after Hezekiah's lifetime) among them.

Secondly, the Old Testament has only a few references to the idea of there being different aspects of God (Psalm 110:1-"YHWH says to my Lord..." is one of the explicit ones, though it implies a relationship without specifically defining it as, say, Father to Son); the emphasis in the Old Testament is on God being 'one' as opposed to many gods. This is best articulated in the Judaic Shema of Deut 6:4-"Hear O Israel, YHWH our God, YHWH is One." The New Testament continues with the idea of God being one god, articulated in the gospels by Jesus (cf Mark 12:29) and in the epistles (Paul, cf 1 Cor. 8:6a); but the New Testament also explicitly references different aspects of the essence of the one God in the Father, Son, and Spirit. Jesus, his disciples, and Paul all speak to this directly throughout the New Testament. The gospels are clear about God being Jesus's Father (cf Matthew 11:27), that Jesus is the Son (cf Mark 14:61-62), that there is God the Spirit (cf Matthew 3:15-17), and that these different aspects have distinct personalities and roles (as in John 14, for example). When you ask which Old Testament verses reference a particular aspect of the triune God, you are taking an explicit definition of Christian tradition ('trinity' does not occur at all in either Old or New Testament; the Trinity is a concept to explain the 1x1x1=1 essence of God portrayed in the Bible), then using it as a measurement against the Old Testament verses that did not concern themselves with which aspect did what. It is effort fraught with futility to ask which Old Testament actions were performed by the Son vs the Father when the text does not denote it. We would be implying intention where there is none.

As to the Son's role articulated in the New Testament, it does say in John 5:27 that God the Father has given the Son authority to judge, but Paul says further in Romans 8:34 that Christ is both judge and intercessor: though the Son passes judgment, he also intercedes and defends humanity. We have no human legal equivalent for this, one person acting as both judge and defending counsel. There is a social/colloquial example of mothers disciplining their children but also extending grace and arguing on their behalf in the face of judgment. Analogy only helps us touch the edges of God's character, though. We should guard against using our experiences as a way of defining who God (any of of his aspects) is or will be. [explanations of the Trinity and references paraphrased from Norman L. Geisler's reference work, "Baker Encyclopedia of Christian Apologetics" entry on 'Trinity'; my references are examples and far from exhaustive. Geisler's work is far more extensive on this.]

I draw your attention to the past tense in:

For the Father judges no one, but has committed all judgment to the Son - John 5:22 KJV (emphasis added)

This said pre-resurrection and glorification; Together with the implications of -

For God sent not his Son into the world to condemn the world; but that the world through him might be saved. - John 3:17 KJV

ie Judgment was not part of Jesus' mission during his 'first coming' (the incarnation); and also:

Jesus Christ the same yesterday, and to day, and for ever. - Hebrews 13:8 KJV

It should be fairly clear that the one who is referred to in:

Because he hath appointed a day, in the which he will judge the world in righteousness by that man whom he hath ordained; whereof he hath given assurance unto all men, in that he hath raised him from the dead. - Acts 17:31 KJV

was, is and will be the judge of all mankind from the very beginning and until the very end of all ages.