The 1835 collection contains no notation whatever. That was ordinary for American hymnals of the period and it shaped how the book was used: a hymn was named, a tune of the right metre was chosen from what the congregation already knew, and the two were joined on the spot. The consequence is that no single melody is authoritative for these texts, and a hymn from this book has often been sung to several different tunes across its life. It also meant the book was cheap to print and easy to carry, which mattered a great deal to a church that kept moving.
The succession splits after Joseph Smith's death in 1844. The larger body that went west with Brigham Young produced its own hymnals in Utah, growing from this root through the nineteenth century and eventually acquiring printed music. Emma Smith did not go west. She remained in Nauvoo, and the church that reorganised around her son in the 1860s, now Community of Christ, traced its own hymnody through her and issued hymnals of its own. Both lines are genuinely descended from the 1835 book, which is unusual: most schisms do not leave the compiler on the smaller side.
Some of its texts have proved durable well beyond either institution. Phelps's hymns in particular, written for a movement that expected the world to end shortly, are still sung weekly by millions of people who do not, and their survival is one of the quieter measures of what a good hymn text can outlast.