I saw a Mormon Web article recently which said that official Mormondom used to blame the "universal apostasy" on the early Christian leaders...but that the Mormon narrative had changed over more to blaming the Christian grassroots.
The current Mormon missionary lesson indeed blames the Christians-at-large back then -- not the leadership.
Ten years ago, a Mormon FReeper had claimed that the term "Christian" came into use after the apostasy had occurred. That places the falling away as occurring during the Acts Of The Apostles (specifically prior to Acts 11:26), since that is where the Bible records Jesus' followers were "first called Christians". If that is true, then the apostasy began less than eight years after Christ's ascension, the LDS' Melchizidek Priesthood was stripped away from the Apostles themselves for the majority of their ministry, and most (if not all) of the New Testament, which was written by the Apostles, was written during their apostasy from Jesus Christ.