What were the Apostolic Fathers's contributions to Christology which survived to Nicene (325 AD) and Chalcedon (451 AD)?

Although they are a diverse group and should not be thought of as presenting a unified view, the Apostolic Fathers contributed to the development of Christology in several ways.

Ignatius of Antioch is perhaps the prime example. His letters were widely read and he implies an extremely high christology in which Jesus is simply called "our God."

• Jesus Christ, our God (Letter to the Ephesians, Chapter 1, Letter to the Romans, Chapter 1).
• Christ our God (Letter to the Smyrnaeans, Chapter 10).
• Our God, Jesus Christ (Letter to the Ephesians, Chapter 18, Letter to Polycarp, Chapter 8).
• God, even Jesus Christ (Letter to the Smyrnaeans, Chapter 10).
• God Himself being manifested in human form (Letter to the Ephesians, Chapter 19).
• God existing in Flesh (Letter to the Ephesians, Chapter 7).

The author of 2 Clement presents a similarly high Christology when he says “Brethren, it is fitting that you should think of Jesus Christ as of God – as the Judge of the living and the dead.” 1 Clement 2, possibly by a different writer, speaks as God as suffering through Christ.

Polycarp of Smyrna also wrote in terms of a high Christology but refrained from referring to Jesus directly as "God":

Now may the God and Father of our Lord Jesus Christ, and the eternal High-priest Himself the Son of God Jesus Christ, build you up in faith and truth, and in all gentleness and in all avoidance of wrath and in forbearance and long-suffering and in patient endurance and in purity; and may He grant unto you a lot and portion among His saints, and to us with you, and to all that are under heaven, who shall believe on our Lord and God Jesus Christ and on His Father that raised him from the dead,” (Polycarp, Letter to the Philippians, Chapter 12).

The Espistle of Barnabas similarly speaks of Christ as the pre-existent agent of creation but not directly as God Himself :

the Lord endured to suffer for our soul, He being Lord of all the world, to whom God said at the foundation of the world, ‘Let us make man after our image, and after our likeness,’” (Epistle of Barnabas, Chapter 5).

Hermas, who wrote the widely read work known as "The Shepherd," made a somewhat clearer distinction between the Father and the Son:

First of all, believe that there is one God who created and finished all things, and made all things out of nothing. He alone is able to contain the whole, but Himself cannot be contained,” (26:1, or Commandment 1).

The Son of God is older than all His creation, so that He was a fellow-councilor with the Father in His work of creation,” (89:2, or Parable 9, Section 12).

In terms of continuity with the NT scripture and later church councils, I will not attempt a detailed analysis. However the high Christology fits well with prologue the Gospel of John and some letters of Paul. The Council of Nicaea would later promote the idea that Christ existed as the Second Person of the Trinity with God the Father from the beginning. It would reject any implication as might be seen in the statements of Hermas that God the Father is greater than God the Son or that God the Father, rather than the Son, was the creator of the heavens and the earth. The councils would further clarify that Christ is not only "our God" but also our "man," being fully human as well as fully divine.

Generally the Apostolic Fathers are most useful to understand early ecclesiology and give a fascinating glance into the life of the early church. But the above excerpts show that several of them also contributed to Christology. These usually present a very high Christology, with Ignatius perhaps veering into what would later be condemned as Monophysitism. Hermas may be an exception to the "high Christology" rule in that he implied the superiority of the Father to the Son. Might he, in a later age, be considered guilty of Arianism? However, in their own day they were all orthodox, and contributed significantly to the emerging proto-orthodox consensus.

Note: I have relied heavily here on the article "The Apostolic Fathers and the Deity of Christ" by Luke Wayne. Thanks to @Mike Borden for providing information from 2 Clement. I am not an expert on this subject but thought the question is a good one and deserves an attempt at an answer.