Prophetic language sometimes condenses timeframes, and while this even appears to follow the coming of the Triumphant Messiah, we know that it actually precedes it, as it is happening today. The “at that day” is simply a general reference to the end times. The gospel is being preached to the “remnant of this people,” whom we know as the Lamanites. The gospel is being preached to the “dispersed of my people” in the various lands into which they were dispersed. The gathering of the children of Israel is underway.