Supposing -- the "how" notwithstanding -- the Neo-Babylonian revolt is averted, and the Assyrian Empire is still (essentially) intact circa 550 BC? Does Cyrus the Great, or someone like him, still emerge in the east around this time? If so, does Assyria fall straight to him? Primary question, how is Middle Eastern history and civilization altered, and what are the butterflies from there?
CONSOLIDATE: Also, I realize a similar
thread was done some years ago...
Well, actually, when the Assyrians fell, they were pretty much at the zenith of their power. Although Egypt broke away during Ashurbanipal's reign, Assyria also destroyed it's old enemy Elam, and Urartu was rendered pretty much caput as well. If a struggle for the succession had not broken out between the sons of Ashurbanipal at exactly the wrong time...when Nabopolassar was consolidating his rule in Babylonia and the Medes were on the rise...Assyria might well have survived.
So let's say the civil war doesn't happen. Assyria remains united and its army...the best in the world...is not weakened by years of civil war following Ashurbanipal's death. Let's say his successor, Ashur-etil-ilani, is a different man than he was in OTL....a strong king in the mold of his fathers who enjoys the complete loyalty of his generals and family. He severely chastises the nomad invasion (an allied horde of Scythians, Medes, Persians, Cimmerians) which hit Assyria at the death of Ashurbanipal, then turns on Nabopolassar, nipping that problem in the bud by defeating, capturing, and flaying alive the Chaldean leader and burning Babylon to the ground, like Sennacherib did, once upon a time. Like his predecessors, he then makes punitive campaigns into Media and Persia, extracting a tribute of horses and mercenary soldiers from the Iranian tribes.
When Ashur-etil-ilani dies in 575 BC, his grandson (lets call him Shalmaneser VI) succeeds him. He too is a man in the mold of Ashurbanipal, and successfully defends the borders of Assyria. He defeats attempts by the Saite kings of Egypt to meddle in Syria-Palestine, and re-establishes Assyrian control over the Nile Delta, including the city of Sais itself (Pharoah Wahibre is forced to move his capital to Thebes). Shalmaneser VI dies in 540 BC, with the Assyrian Empire stronger than it has been since the reign of Esarhaddon.
In this scenario, Cyrus probably does not arise in the East, because Assyria would have been pursuing its policy of frequent raids into Iran to keep the tribes there weak. Assuming Zoroaster lives at all, he probably won't have a strong royal booster for quite some time.
As for the butterflies, Saite Egypt probably lasts a bit longer, assuming the Assyrians don't decide to make an end of it. The empires of the Medes and the Persians either don't arise at all, or arise much later. Lydia in Asia Minor probably continues for some time as an ally and possible client state of Assyria.
There won't be any Persian Wars beginning in 490 BC, and may not be ever. As a knock-on effect, there probably is no Athenian Empire or Golden Age of Athens (in OTL it was the Persian threat which led to the formation of the Delian League, which Athens perverted into the Athenian Empire, stealing the treasury to fund the construction of the Age of Pericles in Athens).
From there, who knows? The Assyrians would probably fall eventually, but the Neo-Assyrian empire had already lasted for several hundred years, and it might have lasted, with luck, for a few hundred more.