See a similar prophecy by Lehi in 1 Nephi 5:18, though Lehi’s oracle is somewhat more limited than Alma’s: Lehi indicates that the brass plates shall eventually go forth to every nation, kindred, tongue, and people who are of his (Lehi’s) seed. How is it that the brass plates shall go forth to all people and nations?
First of all, at least a portion of this sacred record shall go to all the world through the dissemination of the Book of Mormon. Its messages from Zenos, Zenock, Neum, and Ezias, as well as the clarifying and expanding contributions (upon what we call biblical matters) shall come to be known and treasured as the Latter-day Saints flood the earth with the Book of Mormon.
Eventually, however, this prophecy shall be fulfilled in a day yet future, presumably the Millennium, when the brass plates are delivered to the Saints and when God raises up, through his appointed channels, a prophet, seer, and revelator, who will also be a translator (see Bruce R. McConkie, “The Doctrinal Restoration,” in The Joseph Smith Translation: The Restoration of Plain and Precious Truths, p. 16). Then the contents of this remarkable volume will be unfolded to all mankind.