Before we begin to see whether there is a conflict with the old testament scriptures* which says that Yahweh is the judge even of eschatological issues and with the new testament which attributes all eschatological judgements to Christ, it is first wise to understand who Jesus Christ is.
Jesus Christ in the gospel of John identifies himself to be Yahweh of the Old testament when in John chapter 8:24 and 58 he clearly says “If you do not believe that I AM you will die in your sins… Before Abraham was born I AM”. Jesus here identifies himself to be the Yahweh of the Old testament, that he is the God of Abraham, Isaac and Jacob. When Jesus identified himself as Yahweh the Jews wanted to kill him because they did not know that He was telling the truth. Thus in the New testament we see a clear revelation of Jesus Christ as the Son of God and because He is God he is able to judge perfectly. See 2 Corinthians 5:10 1Corinthians 4:5 cf. Jeremiah 17:10
Therefore there is no conflict between the Old testament and the new testament. Yahweh is assigned the role of Judge and because Jesus is Yahweh he is very capable to fulfil this duty. The new testament brings about a fuller revelation of who God is. That is that God is Father, Son and Spirit and although there are allusions to this in the old testament the New testament displays the full reality. See 1 Corinthians 4:5 Hebrews 1
But what is the Father’s role? Jesus came to make known the will of the Father, and the will of the Father was for Jesus to have authority to judge because Jesus is the son of man (John 5:27) which is a reference to Daniel 7:13-14. Also the Father’s will is that any one who looks to and believes in Jesus will have eternal life and Jesus will raise them on the last day (John 6:39).
But the Judgment of Jesus is not merely his own Judgment but he works in unison with His Father. “By myself I can do nothing; I judge only as I hear, and my judgment is Just, for I seek not to please myself but the one who sent me”, John 5:30.
The Father has entrusted all judgement to Jesus Christ His Son and the Father’s role I believe is that He is working in unison with His Son and while the Father will not be doing the Judging in a physical sense yet His will is being enacted because whatever judgement Jesus passes, it is as if the Father passes it also because they are “One”. See Romans 2:16
Jesus is the Kyrios and the Father has placed all things into his hand (Matthew 28:18) so that all may worship and honour Jesus just as they honour the Father. The Father appointed Jesus to be the judge of all. This is a revelation. See Acts 10:42 Acts 17:30-31
* 1 Samuel 2:10 Psalm 94:1-2 Psalm 7:6-8 Isaiah 66:15-16 Joel 3:1-2, 11-17 Isaiah 33:22
This has not been dealt with exhaustively so please asks questions and I will react to your questions.