The prophets were prophesying that Babylon would destroy Jerusalem, but they were probably not believed because Babylon had already dominated Judah and Jerusalem. Zedekiah was king only because the Babylonians installed him. Thus, the prophets were not saying that Babylon would come, because it had already come. They were prophesying an escalation of contention that perhaps was more difficult to see in the year they installed Zedekiah as king.
The first year of the reign of Zedekiah gives us a starting year for the Book of Mormon. It was 597 BC. Jerusalem was destroyed about 10 years later, and Lehi and his family left prior to the destruction, but perhaps not too long before the destruction.
Perhaps ironically, perhaps symbolically, it was about one hundred years earlier that Assyria and invaded Judah and threatened Jerusalem. Prior to both invasions, a king in Judah had attempted to reform Israelite religion, with only temporary success. The people began returning to their previous ideas as soon as the next king was anointed.
These parallel invasions undoubtedly helped convince Nephi to see his family, and later his people, in the context of a scattered people who were to be gathered to Israel. Isaiah was the prophet of the Assyrian invasion, and the scattering and gathering of the ten tribes were important to Isaiah. Nephi certainly likened his own people to Isaiah’s times, and saw Isaiah as particularly poignant for his New World people.
More immediately, however, the fact that both the Assyrians and Babylonians had invaded Judah, but Jerusalem yet stood—must have informed the popular opinion, and provided what must have been seen as the logical argument against the doomsaying prophets,among whom was Lehi.