How is it possible to trace the beginning of Jesus' divinity chronologically?

Question

Saint Ignatius was bishop of Antioch in Syria between 68 and 100 or 107, a disciple of the apostle John. He also knew Saint Paul and was the successor of Saint Peter in the church in Antioch. According to Eusebius of Caesarea, Ignatius was the third bishop of Antioch in Syria and according to Origen he would have been the second bishop of the city. Saint Ignatius was detained by authorities and transported to Rome, where he was sentenced to death in the Colosseum martyred by lions.

Epistle of Ignatius to the Ephesians, chapter 18:

For our God, Jesus Christ, was, according to God's designation, conceived in the womb by Mary, of the seed of David, but by the Holy Spirit.

Approximately at this same time, 110 AD and in nearby regions, we have the main historical document of the existence of a Christ worshiped as "god", written by the enemies of the Church, which makes it even more valuable and pertinent: Pliny the Younger, on Christians, a letter to Emperor Trajan.

... There were among them those who assured, moreover, that their entire fault, or rather, their entire error, consisted solely in meeting regularly on a certain day before sunrise, alternately singing among those present a hymn in honor of Christ, as if he were a god...

The term Θεὸς "God" can also refer to the term "Elohim" which encompasses the meanings of human magistrate, the divine Christ and also Almighty God himself. The two citations to the previous narratives refer only to the uncircumcised church outside Judea (Acts 20:20-27).

In the letter to the Romans 6.1, Ignatius of Antioch writes:

The ends of the world, nor the kingdoms of this age, will be of no use to me. It is more wonderful for me to die for Jesus Christ than to reign to the ends of the earth. It is Him I seek, who died for us; I want Him who was resurrected because of us. My birth awaits me. 2 Forgive me, brothers: do not want to stop me from living, do not want me to die; what wants to be from God, do not present it to the world nor seduce it with matter. Allow me to receive pure light: when I get there I will be a man. 3 Allow me to imitate the suffering of my God. If someone has it within themselves, they will know what I want and will sympathize with me, because they know what drives me.

In the Epistle to the Smyrniots, chapter 1 says:

I give glory to Jesus Christ, the God who granted him such wisdom

What reasons led the recent Church of the uncircumcision and Ignatius to treat Jesus as a God?

Answer 1

Many Christians answer this question by showing that belief in the Deity of Christ was not introduced circa the time of Ignatius of Antioch, but goes all the way back to the beginning of the Christian era.

A few examples:

and they shall call his name Emmanuel, which being interpreted is, God with us (Matthew 1:23b)

But unto the Son he saith, Thy throne, O God, is for ever and ever: a sceptre of righteousness is the sceptre of thy kingdom. (Hebrews 1:8)

In the beginning was the Word, and the Word was with God, and the Word was God (John 1:1).

Jesus said unto them, Verily, verily, I say unto you, Before Abraham was, I am (John 8:58).

And Thomas answered and said unto him, My Lord and my God. (John 20:28)

There are Christians who understand these verses differently and they might offer a very different response to the OP's question. But for Christians who see that the Deity of Christ is plainly taught in the New Testament, the answer to the OP's question would be:

Q: What reasons led the recent Church of the uncircumcision and Ignatius to treat Jesus as a God?

A: Because that is what the apostles taught.

Answer 2

All of the people you mention were founding their faith on the very words of God in the Hebrew scriptures and on the heard or reported words of Jesus of Nazareth and on the heard, read or reported words of the apostles whom Jesus sent.

Within that context it is clear that the reason for anyone, at all, believing that Jesus of Nazareth is the Son of God is a matter of personal revelation.