The first and most obvious impression is that since Jesus had once referred to His disciples as "brothers," the significance of πατέρα μου καὶ πατέρα ὑμῶν would therefore be to their mutual God and Father as "brothers."
John 20:17 (NASB)
17 Jesus said to her, “Stop clinging to Me, for I have not yet ascended to the Father; but go to My brethren and say to them, ‘I ascend to My Father and your Father, and My God and your God.’”
However, there is more nuance at hand in the writings of John.
For example, the only reference in John to πατέρα μου / πατέρα ὑμῶν and θεόν μου / θεὸν ὑμῶν is John Chapter 8. In that passage, Jesus makes the contradistinction between His Father and God, who is truth, and Satan, who was a liar from the beginning and who was therefore the father and god of the Pharisees, who had denied the testimony of Jesus.
John 8:41-44 (NASB)
41 You are doing the deeds of your father.” They said to Him, “We were not born of fornication; we have one Father: God.” 42 Jesus said to them, “If God were your Father, you would love Me, for I proceeded forth and have come from God, for I have not even come on My own initiative, but He sent Me. 43 Why do you not understand what I am saying? It is because you cannot hear My word. 44 You are of your father the devil, and you want to do the desires of your father. He was a murderer from the beginning, and does not stand in the truth because there is no truth in him. Whenever he speaks a lie, he speaks from his own nature, for he is a liar and the father of lies.
The Apostle John later says that the liar is the one who denies Jesus Christ, and therefore the Father.
1 John 2:22-23 (NASB)
22 Who is the liar but the one who denies that Jesus is the Christ? This is the antichrist, the one who denies the Father and the Son. 23 Whoever denies the Son does not have the Father; the one who confesses the Son has the Father also.
Shortly before his arrest, the disciples had promised to be faithful to Jesus (Matt 26:35), but later they fled from Jesus in the Garden of Gethsemane. The disciples thus broke their word to remain with Him. When Peter was then confronted about his association with Jesus, had lied three times: that is, he denied ever knowing Jesus.
In conclusion, the first impression of John 20:17 is that the disciples are the "brethren" of Jesus and thus the choice of words "my Father / your Father and my God / your God. However, their broken promise to Jesus to be faithful made them "liars." (The "father" of lies is the devil, and therefore he is the "god" of those who deny the testimony of Jesus.) Peter too lied about Jesus before several people when he was confronted. When Jesus therefore used the phase, πατέρα μου καὶ πατέρα ὑμῶν, he had eliminated any doubt in the minds of the disciples that their "father" was none other than the Father in heaven, who is also the Father of Jesus.