What does the Anglican Church teach about being born again?

The Anglican Church in North America (ACNA) is the name of a denomination formed in 2009 initially consisting generally of people seceding from the Episcopal Church of the USA and the Anglican Church of Canada. The reason for the secession was concern that the ECUSA etc was becoming too liberal, incorporating un-Biblical and un-Anglican doctrine and practice. Same sex marriage was one example.

ACNA includes both evangelicals and Anglo-Catholics. As Nigel J points out there is a great diversity even in traditional Anglicanism. To an extent, this reflects a particular approach to sola scriptura. The Bible says whatever it says, and the Anglican approach shies away from adding to this any particular interpretation or systematic theology of its own. Historically, the Church of England, on which Anglicanism is based, rejected the authority of the Pope but without claiming any such authority itself.

Anglican doctrinal statements are often said to be deliberately vague and ambiguous. Some say this is an attempt to please everybody. Alternatively, it can be seen as a recognition that where the Bible can reasonably be interpreted in multiple ways, the Church has no right to go beyond this.

The 1662 Book of Common Prayer is considered doctrinally definitive for most Anglicans. It contains, in the adult Baptism service, the following exhortation based on the Nicodemus passage quoted by OP.

BELOVED, ye hear in this Gospel the express words of our Saviour Christ, that except a man be born of water and of the Spirit, he cannot enter into the kingdom of God. Whereby ye may perceive the great necessity of this Sacrament, where it may be had. Likewise, immediately before his ascension into heaven, (as we read in the last Chapter of Saint Mark’s Gospel,) he gave command to his disciples, saying, Go ye into all the world, and preach the Gospel to every creature. He that believeth and is baptized shall be saved; but he that believeth not shall be damned. Which also sheweth unto us the great benefit we reap thereby

This seems to connect the phrase with Baptism. But what about the case of a person who believes but is not baptized? Jesus did not say and neither, officially, does Anglicanism presume to do so either.

The Anglican church of North America has published its own catechism catechism

In it is a modernised version of the Collect for Christmas Day

Almighty God, you have given your only-begotten Son to take our nature upon him, and to be born of a pure virgin: Grant that we, who have been born again and made your children by adoption and grace, may daily be renewed by your Holy Spirit; through our Lord Jesus Christ, to whom with you and the same Spirit be honor and glory, now and forever. Amen.

The phrase "born again" replaces the word "regenerate" in the original collect. This suggests they are seen as synonyms.

The ACNA catechism also contains the following section headed TURNING TO CHRIST.

Turning to Christ brings us into fellowship with God. Baptism, which is the rite of entry into the Church’s fellowship, marks the beginning of this new life in Christ. The apostle Peter, proclaiming the Gospel, said, “Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit” (Acts 2:38). the gospel. Through faith, repentance, and Baptism we are spiritually united to Jesus and become children of God the Father. Jesus said: “I am the way, and the truth, and the life. No one comes to the Father except through me” ( John 14:6). As we come to the Father through Jesus Christ, God the Holy Spirit enlightens our minds and hearts to know him, and we are born again spiritually to new life. To continue to live faithfully as Christians, we must rely upon the power and gifts which the Holy Spirit gives to God’s people.