Because we have already seen that Nephi sees that the sealed portion itself is t be reserved in time, it is the current text of the Book of Mormon that is the fulfillment of this particular prophecy. The translation of the book comes through the power of the Lord, and therefore "they shall be read by the power of Christ." The missionary work to spread the gospel is the fulfillment that the words "[they] shall be read upon the house tops."
The last sentence may contradict this reading, as it deals with revelation of things past and future. However, in the prophetic mode of collapsing time, this may simply be a later phase of the prophecy, or may have a parallel fulfillment in the return of living prophets to the earth.