Let us tread very carefully here as there are minefields to the left and right. However, I congratulate the OP on noticing a very significant connection between these important titles.
It is true that John ascribes the unusual title, "ὁ ὢν καὶ ὁ ἦν καὶ ὁ ἐρχόμενος" (or a slight variant) as follows:
- Rev 1:4 - Grace and peace to you from him who is, and who was, and who is to come
- Rev 1:8 - "I am the Alpha and the Omega," says the Lord God, "who is, and who was, and who is to come, the Almighty."
- Rev 4:8 - ... "Holy, holy, holy is the Lord God Almighty, who was, and is, and is to come."
- Rev 11:17 - Lord God Almighty, the One who is and who was, because you have taken your great power and have begun to reign. [Note the lack of "the one to come because at this point, God as already come.]
However, we have a pretender to and would-be usurper to God's position who is given a similar title - the great red beast, the dragon from the abyss:
- Rev 17:8 - The inhabitants of the earth whose names have not been written in the book of life from the creation of the world will be astonished when they see the beast, because it once was, now is not, and yet will come.
- Rev 17:11 - The beast who once was, and now is not, is an eighth king. He belongs to the seven and is going to his destruction.
As cleverly pointed out by the OP, these same verbs, "was, is, to come" are used of Jesus in the Gospel of John, although not in the same sentence but spread over the early chapters. Thus, they are never used as a title of Jesus but subtly woven into the narrative.
- ἦν (was) - John 1:1 "in the beginning was the Word"; John 1:15 "[the Word] was before me"
- ὢν (is) - John 1:18 "is in the bosom of the Father"
- ἐρχόμενος (to come) - John 1:15 "[the Word] come after me", John 1:27 "the one coming after me", etc.
Nothing so ruins a good case as when it is over-sated (according to Greek Rhetoricians). While the title, "the one who is and who was and is to come" is definitely a title of the Father as listed above, it is never used as a title of Jesus.
However, the closest we come is John 1:15:
John testified concerning him. He cried out, saying, "This is the one
I spoke about when I said, 'He who comes after me has surpassed me
because he was before me.'"
Again, this is not a title but a statement of historical fact that Jesus existed before John the Baptist.
Lastly, the Title of the Father, "the One who was, and is, and is to come" is also a statement of fact that the Father is eternal (as well as a title) - He has always existed, and always will exist. The John goes to some length to assert precisely the same thing about Jesus, the Word, namely that:
- Jesus has always existed with the Father (John 1:1, 15, 8:58)
- Jesus does exist (John 1:18)
- Jesus will always exist (see Rev 11:15 - "he will reign forever")
Thus, while the eternity title, "the one who was and is and is to come" is applied exclusively to the Father and never to the Son, the Bible still asserts the eternal existence of the Son emphatically.
APPENDIX - Other Scripture on the same topic
In the Old Testament, Jehovah’s self-proclaimed title of “I AM” is given special prominence in Ex 3:13-15. It appears to suggest the eternal, self-existent one as a title of YHWH. The unpredicated phrase, “ego eimi” (= I am), occurs in the LXX in a number places (Deut 32:39, Isa 41:4, 43:10, 13, 25, 45:19, 46:4, 48:12, 51:12, 52:6) and always refers the One and Only Great Jehovah God Almighty.
John uses this unpredicated "I AM" in his Gospel as follows:
- John 4:26 – “Then Jesus said, ‘I am.’” [To the Samaritan woman at the well. There is a reasonable case for this being identification, but that is a matter of taste.]
- John 6:20 – “But then [Jesus] said to them, ‘I am. Fear not.’” [To the frightened disciples in the boat.]
- John 8:24 – “If you do not trust/believe that I am, you will die in your sins.”
- John 8:28 – “When you will lift up the Son of Man, then you will trust/know that I am.”
- John 8:58 – “Truly, truly, I say to you; before Abraham existed, I am.” [The Jews then tried to stone Him for blasphemy.] Note that this and the previous two mean that Jesus, in the space of this chapter of John 8 uses the unpredicated “I am” idea in the present (v24), future (v28) and past sense (v58). V24 & 28 appears to be tied to believers’ salvation as well.
- John 13:19 – “From now [on] I tell you before the occurrence, that you may believe when it occurs that, I am.”
- John 18: 5, 6, 8 – “He said to them, ‘I am.’ …Therefore, when He told them, ‘I am’, they fell backward to the ground.” [This occurred when the Jews tried to arrest Jesus in the garden. It could be reasonably argued that this is a case of identification. However, the fact that the arresting mob fell backward suggests that much more is intended here.]