I was recently engaging with an LDS user on this site who said the true church fell away very early. How early was this? Does the church have official teaching on it and why did it happen?
Priesthood keys were restored to the earth in the 19th century through the Prophet Joseph Smith. Of particular note are the keys & authority given to Joseph Smith & Oliver Cowdery by Peter, James, and John (who appeared as heavenly messengers) in June of 1829:
12 And also with Peter, and James, and John, whom I have sent unto you, by whom I have ordained you and confirmed you to be apostles, and especial witnesses of my name, and bear the keys of your ministry and of the same things which I revealed unto them (Doctrine & Covenants 29:12)
Unofficial Answer from a history nerd
Apostasy is a process, not an event, but we can estimate the dates of several critical junctures in that process.
The apostles held priesthood keys, which are necessary to administer Christ's church. The Church of Jesus Christ of Latter-day Saints does not believe in apostolic succession, but rather, that those keys were taken from the earth at the end of the apostolic ministry, and restored in the 19th century through the prophet Joseph Smith, as discussed above.
When was the end of the apostolic ministry?
Multiple early sources--most notably Irenaeus of Lyons, grandson in the faith of John the Beloved--report that John ministered in/around Ephesus into the reign of Trajan, who began to reign in AD 98 (see Against Heresies 2.22.5). That puts John in active ministry well beyond any estimated date for any of the other apostles. Per Jerome, we can estimate the end of John's ministry early in Trajan's reign, approx. AD 100. When Ignatius wrote to the Ephesians (circa AD 107), John was not mentioned and it is quite apparent he was no longer among them by that time.
Estimate for the end of John's ministry: AD 100. By this point, the church is no longer led by apostolic keys.
As Tad R. Callister1 has pointed out in The Inevitable Apostasy (see pp. 52-53), for several decades thereafter, there would still be church leaders officiating in capacities to which they were ordained by the apostles. Individual Bishops would not have had authority over the entire church--Ignatius himself is explicit that his authority is inferior to that of the apostles (see Epistle to the Romans ch. 4)--but they retained authority granted to them by the apostles to oversee their part of the flock.
The last known individual leading a portion of the church with authority given to him by the apostles is Polycarp of Smyrna, who died circa AD 155. After this time, the church was neither led generally by apostolic keys, nor locally by people authorized by the apostles.
I propose AD 100 & AD 155 as two of the more significant points along the spectrum from a fully-functional, authorized church, to a state where the authority was no longer present.
Notes1 Callister's work is one of the more thorough reviews of the Patristic writings showing evidence of the growing apostasy of the early church
2 Claims that the early church fell into apostasy and lost authority are understandably disagreeable to many. My intent is not to give offense, but to acknowledge what we believe.
While I believe the fulness of the Gospel of Jesus Christ was no longer on the earth during the Great Apostasy, I believe there was much of His truth and His Gospel influencing the world. Countless genuine, faithful Christians lived and died during this time, faithfully following the light & truth that they had.
It is my studied opinion that no document has had a greater impact on world history than the Gospel of Matthew, the foundational text of the Christian movement. While the Gospel of Matthew does not contain everything Jesus taught, it led the way in taking faith in the Savior Jesus Christ to the ends of the earth.
The Church of Jesus Christ of Latter-day Saints does not have an exact date when the true church ceased to exist. It was a gradual process over years.
After the deaths of the Savior and His Apostles, men corrupted the principles of the gospel and made unauthorized changes in Church organization and priesthood ordinances. Because of this widespread apostasy, the Lord withdrew the authority of the priesthood from the earth.1
See also The Great Apostasy
That is a very hard question as history is written by the victorious. To find the truth one must dig very deep. The Romans had a religion. Jesus had teachings. Mix those two, you get the Roman church (like Christmas tree, Saturnalia, Christmas date, son of God, Trinity, human woman mother of the son of God, etc.).
Romans held the Council of Nicaea, decided what the right faith is, and what the Bible is. The Roman/Church wiped all others. The winners are the Romans.
This is a very short answer. While from my research I found that Arianism is the closest to the original of Jesus' teachings, you are welcome to do your research.