Isaiah is a complicated book to follow if you're not going with uncritical/apologetic theology. Understanding what exactly is going on requires a fair bit of linguistic and paleographic analysis. The simple version is that scholars view the single canonical Book of Isaiah as at least three separate works from different points in time. They're labeled Proto-Isaiah, Deutero-Isaiah, and Trito-Isaiah (Greek for first, second, third). The writing style in each section is markedly different from the other two, indicating separate authors. Likewise the Hebrew used in Proto-Isaiah is more archaic than that of the other two, indicating a substantially earlier date.

Proto-Isaiah covers chapter 1-39 in modern Bibles, and is generally accepted as the work of a historical or prophet (or his scribe) working in Judah during the late 8th Century CE, just as the author claims. It frames large sections of the text as prophetic visions of the future. Some have to do with more abstract theological futures, others are concerned with the politics of the wider region. Chapter 24 is even some of the oldest proto-apocalyptic literature we know of.

Deutero-Isaiah covers chapters 40-55 clearly shifts context. It stops mentioning the author by name and actually is not framed as prophecy. This first third or so is almost entirely framed in the future and past tense. It also refers to the Persian king, Cyrus the Great, by name and show much more familiarity with Babylonian/Mesopotamian institutions than pre-exilic writing. Even the most ambitious predictive prophecies in the Bible tend not to include the finer details like that, hinting at near-contemporary authorship. The academic consensus is that Deutero-Isaiah was writing during the Exile in Babylon, more than a century after the original Isaiah's time. Many of the future tense passages are framed as things that God had previously said to other people, but recorded after the fact. It's not framed as the author's own pre-knowledge.

Trito-Isaiah covers the rest of the book, 56-66. This section remains anonymous, but returns to the predictive format of the original Isaiah. The writing style is not as consistent as the other two chapters, and each of the oracles in Trito-Isaiah is fairly disconnected from the message of the others, so they may be the work of many authors. Unlike either of the preceding sections, they are not concerned with the impending doom of imperial conquerors and the struggles of the exile. Instead, they are concerned with theological questions and political issues facing the renewed province of Judea in the Early Second Temple period, after the Exile. Several of them also represent some of the earliest Messianic predictions in the Bible.

It should be noted that Isaiah 44 is where a long section partially framed in the future-tense does start, but it is not framed as a prediction in the far off past. Chapter 44 actually has very little to do with the Fall of Babylon, which is more the provenance of Chapters 45-47. Chapter 44 on the other hand is a prediction about the joys the Jews would experience when they returned from their Exile. The Return is also the focus from Chapter 48-55, which are likewsie fairly vague about precise details, largely concerned with how good it will be and how they will restore Jerusalem (goals that were well established in the exile community. Chapters 45-47 present themselves as contemporaneous with the Persian invasion of Babylonia, with God speaking to the author or the Exile community at large.

Thus says the Lord to his anointed, to Cyrus, whose right hand I have grasped to subdue nations before him and to strip kings of their robes, to open doors before him—and the gates shall not be closed: (45:1 is God speaking in the present about something that shall happen)

I have aroused Cyrus in righteousness, and I will make all his paths straight; he shall build my city and set my exiles free, not for price or reward, (45:13 is part of the same speech, but God has already aroused Cyrus.)

Assemble yourselves and come together; draw near, you survivors of the nations! They have no knowledge—those who carry about their wooden idols and keep on praying to a god that cannot save. (45:20 starts God's instructions for what to do next as Cyrus approaches. This continues through all of Chapter 46 and switches to God speaking to Babylon in 47.)

Even if we unreasonably assume that Deutero-Isaiah was composed entirely in the moment when it is framed, and that the author never edited it after the fact, predicting that Cyrus would conquer Babylon and follow the policies he had used in more than a decade of conquests is hardly a stretch.

Interestingly though, chapters 44-55 are not the only predictions that Babylon would fall in the canonical book. Proto-Isaiah actually predicts it as well, twice, in Chapters 13 and 21. That's not terribly remarkable. Proto-Isaiah predicts the downfall of basically everybody:

Israel, Assyria, Babylon, the king of Babylon, Assyria again, Philistia, Moab, Damascus, Nubia, Egypt, Assyria and Israel again, Egypt and Nubia again, Babylon again, the Edomites, Arabia, Judah itself, the city of Tyre, the entire world, the leaders of the world, Judah and Egypt again, and last but not least, everywhere except Judah.

The trick is that Babylon (and all the rest of these frankly) was conquered many time in ancient history. Proto-Isaiah was writing during the height of the Assyrian Empire, which faced rebellion and resistance from its vassals and conquered new territory constantly. Predicting that armies from Assyrian subject peoples would one day destroy any given kingdom in the region was basically stating fact. However, both predictions about Babylon in particular draw a lot of attention from apologists because they mention the Medes, the dominant power in Iran before the Persians, whose name is often used in place of "Persian" in Classical literature.

Chapter 13's prediction against Babylon has very little in common with the largely peaceful occupation of the city by Cyrus the Great. It describes a violent conflagration and sack of the city compared to the destruction of Sodom and Gamorrah. Chapter 21 gets closer, describing a Babylonian coming from northern Arabia only to fall in a conflict involving Elam and Media (Elam being the historical region that became Persia). It also describes the occupiers taking over the city in relative comfort rather than wholsale destruction.

If you only know the big stories of Babylonian history, it's easy for this to sound like the Babylonian Nabonidus falling to Cyrus the Great, but devil is in the details here. The same wrath that falls on Babylon in Chapter 21 continues west back to northern Arabia and Edom, which is not something reflected in any record of the Persian conquest. Elam is only referenced as a particpant, not as part of the Medes sacking Babylon.

In reality, the closer fit to the events described in Isaiah 21 is the rebellion and defeat of the Babylonian king Marduk-apla-iddina, also known in 2 Kings and Isaiah 39 as Merodach-Baladan. He allied with Elam, Aram-Damascus, and Israel to rebel against the Assyrian king Sargon II in 722 BCE, fled to Arabia after a defeat in 710, sought aid from King Hezekiah in Judah and was refused, and briefly returned to Babylon in 702 only to be defeated and chased back to northern Arabia. These events are documented in Assyrian chronicles, other parts of the Bible, and Marduk-apla-iddina's own monuments, and make the prophecy in Chapter 21 a contemporary event in Isaiah's own life, referenced in Chapter 39, rather than something a 150 years later.

Edit: I also clicked on your profile and saw your question about reading Babylonian sources that got picked up by the no-question-mark filter. I recommend James B. Pritchard's Ancient Near Eastern Texts Relating to the Old Testament (ANET).