The answer to this question is quite different if you’re asking what we would know about Israel and Judah if the Bible was never written, versus what we would know about Israel and Judah if the books of the Bible were written at one point, but have since been lost. Since your question is specifically regarding the history of Israel and Judah as presented in the Bible, I will not be discussing the history of Israel following the destruction of the Second Temple in AD 70, as the Bible is already silent on Israel’s history following the Temple’s destruction. I am also going to assume your definition of “Bible” is a variation of the Christian Bible (containing the Old and New Testaments).

Archaeological evidence already gives us a rather vivid picture on the development of Ancient Israelites’ religion (statues from pre-exilic times (before the sixth century BC) imply that the Israelites were polytheist or henotheist (the statues depict what appear to be gods (such as YHWH) and goddesses (such as Asherah), while the sudden absence of statues following the Babylonian Exile (587 BC) demonstrate a rather quick change on how the Israelites viewed idols. Biblical texts appear to change how they describe the worship of YHWH overtime (compare YHWH speaking in a plural manner in Genesis “[Adam] has become like one of us, knowing good and evil” (Genesis 3:22, NRSV), to YHWH telling the “gods” they will eventually die but He will live forever “‘You are gods, children of the Most High [...] nevertheless, you shall die like mortals, and fall like any prince” (Psalm 82:6-7), to the Israelites being told to primarily worship YHWH “you shall have no other gods before me” (Exodus 20:3), to Isaiah denying the existence of other gods altogether “I am the LORD [...] besides me there is no god” (Isaiah 45:5). The change in tone as to how Biblical authors viewed their potential Ancient Israelite pantheon helps support the theory that the sudden disappearance of statues following the Babylonian Exile perhaps coincides largely with the shift from polytheism (worshiping multiple gods) to henotheism (only worshiping one god while not denying the existence of other gods) to monotheism (believing in only one god). Without access to the Biblical attitude of the gods, however, it could be more difficult for us to determine approximately when the Israelites became monotheist (this could perhaps be easier to determine if we would hypothetically know contemporary Jews are monotheist, otherwise, the sudden forgoing of idols would, at the very least, tell us the Israelites largely changed their views on idols in some way around the sixth century BC. I am not as well versed in archaeological evidence as I am in textual evidence, so I will try to focus more on what information we would know about the Israelites thanks to the writings of non-Biblical authors.

I am first going to approach this question through the lens of the Bible existing at one point, but becoming lost overtime, let’s assume that all other ancient texts (so long as they are not direct commentary on the Bible) have survived. In this case, we still have quite the wealth of knowledge of Israel’s history thanks to one very important source: Flavius Josephus. Josephus was a Jewish historian active in the late first century AD who fought against the Romans in the First Jewish-Roman War, before eventually surrendering to the Emperor Vespasian. Josephus spent his remaining career documenting the war in his rather appropriately titled work The Jewish War. Though the first chapters of The Jewish War provide background information regarding the centuries before the First Jewish-Roman War (the narrative starting at the outbreak of the Maccabean Revolt c. 167 BC, and the Jewish-Roman War starting AD 66), Josephus states in the preface to his Greek translation that writing a history of the Jews during Biblical times would be redundant due to Greek translations of the Bible such as the Septuagint “An account of the early history of the Jews, their origins, their exodus from Egypt [...] would, I think, be out of place here, and in any case unnecessary; for many Jews before me have accurately recorded the doings of our ancestors, and their accounts have been translated into Greek with very few mistakes”. Luckily for us, should the Bible somehow become lost, Josephus later changed his mind, and had written a later, much more extensive text following the publication of The Jewish War called Antiquities of the Jews. Antiquities, contrary to the Greek preface of The Jewish War, is Josephus’ account of the history of the Jews all the way from creation (from the Biblical perspective) to the First Jewish-Roman War. Josephus’ description of Jewish history very closely parallels the Biblical account in both scope and detail. Josephus’ Antiquities is meant to educate a Greek audience that would have never read the Bible, Josephus makes sure to not just present the history of the Jews, but to also present Jewish law (the latter portion of Book III contains a retelling of many of the Mosaic laws found in Exodus and Leviticus). If our knowledge of the Bible were to disappear tomorrow, little would change regarding our understanding of Israelite history so long as Josephus’ Antiquities survive.

So, first thing I'll do is list a couple of pertinent sources that are ancient Near Eastern history sources. These sources will in many ways account for the biblical text, but they treat the biblical text generally no different from any other ANE textual data. (I'm linking Amazon but please don't order from them.)

The best treatment of ANE history in a comprehensive manner is A. Kuhrt's two volume ANE History.

Van de Mieroop has a more digestible, albeit less thorough single volume history.

Now, to your question.

I should point out that u/Kirbyfan107 has misrepresented a number of things.

This notion of "statues" this user mentions is misleading and minimally helpful. Misleading in the sense that the Asherah was a pole and not a statue. We can look to the inscriptions from Kuntillet Ajrud for textual data on this. There may also be iconographic data as well, but it is difficult (even impossible to some) to determine whether the iconography has any relationship at all with the text of the inscriptions. But, ultimately, as this user admits, they're looking at textual data that is generally quite far removed from the period under discussion. Thus, we have to look at the archaeology first and foremost.

The basic implication of your question is generally correct. I.e., without the biblical text, there is a great deal of information we might not otherwise know. We do not have any massive, monumental inscriptions hailing from the "biblical period" that list kings and their reigns and the deeds that they did. We have no extra-biblical evidence for the Exodus. We also lack any direct evidence supporting the period of the patriarchs. So, we have to look elsewhere: the archaeological and epigraphic records.

The scope in question here is generally ~1200–586 BCE. Eric Cline's 1177 BC details what happened...well...in 1177. Basically, the entire geo-political system in the eastern Mediterranean collapsed, with the light exceptions of Egypt and "Phoenicia" (scare quotes here because Phoenicia wasn't actually a thing...there were about 5 main cities that comprised "Phoenicia" [e.g., Tyre, Sidon, Byblos...I always blank on the other two]). There is very little data we have recovered to be able to say much between 1200–980 BCE or so. The Iron I period (in keeping with Amihai Mazar's Modified Conventional Chronology [get out of here with any of that low chronology stuff that Finkelstein started in 1996—he's even rolling back his own stance on that these days!]) is generally around 1200–1000 or maybe as late as 980. The Iron I, sadly, has little material evidence to support much of anything. What we do have are a handful of fortified sites like Qeiyafa and Lachish. Heavy fortifications require substantial amounts of human capital to be able to construct. So, there was certainly some kind of system in place to plan and execute building operations of such a magnitude. (I'd add, if we dip briefly back into the last bit of the Late Bronze age, the Central Highlands in what is now Israel consisted of a loose confederation of city states that bear no evidence that would support the texts of Exodus or the Conquest in Joshua—the settlement patterns and numbers just don't work.) Additionally, the archaeological reality of Jericho suggests that there was no collapsing of walls or fortifications that we can date to a period that would fit the narrative. The situation is similar at Ai, where the "ruins" (ʿay in Hebrew) date to much earlier than the conquest's purported chronological horizon.

We do know that there was a substantial amount of copper mining at Khirbet en-Naḥash at the southern tip of Jordan around the first quarter of the 10th c. BCE (Thomas Levy, et al., “Reassessing the Chronology of Biblical Edom: New Excavations and 14C dates from Khirbat en-Nahas (Jordan),” Antiquity 78 (2004): 865–79). This means that there was likely also some kind of sophisticated (or at least, sophisticated enough) political and scribal apparatus to control it. (I.e., copper is a money maker and nothing drives social development quite like money.) It very well could have been a Davidic monarchy controlling those resources. (For this theory see Holladay Jr., John S. and Stanley Klassen. “From Bandit to King: David’s Time in the Negev and the Transformation of a Tribal Entity into a Nation State.” Pages 31–46 in Unearthing the Wilderness: Studies on the History and Archaeology of the Negev and Edom in the Iron Age. Edited by Juan Manuel Tebes. Leuven: Peeters, 2014.)

We also know that there were several sites in use during the Iron II period (and even earlier), as I mentioned above, that are mentioned in the biblical text: Lachish, Arad, Gezer, Bethel, Bethlehem, etc. At many of these sites in the Central Highlands or even just nearby, we have found that there appears to be a generally consistent "cultural cognitive code" that led to the production of a consistent type of pottery (including the collared rim storage jar), a consistent kind of domestic architecture (the four-room house), and a general lack of zoological remains (especially pig bones).

The earliest things we can point to that support certain (albeit small) historical data are things like the Tel Dan inscription. This inscription is monumental in nature and is written in the Old Aramaic language. It is also the only extra-biblical reference we have to David. Unfortunately, the inscription is fragmentary, but we have a very clear בית דוד (= "House of David"). Some scholars, of the Sheffield and Copenhagen schools, are inclined not to read this in the simplest manner because they are belong to what we've come to term as the "minimalist school." (Baruch Halpern dismissed their concerns in the 90s with a magisterial article dismantling their arguments, so I'll leave that to the side for now.)

Other things we can point to that speak directly to the historicity of the biblical text are things like the Siloam Tunnel in Jerusalem and the Taylor Prism. The Syro-Ephraimite War of 734 (really rolls off the tongue) points to the historical verisimilitude of Isaiah 7, where Rezin and Pekah attempt to pressure Ahaz into a not so great alliance. We can also look to references like Egypt being described as a "broken staff" later in Isaiah are pretty legitimate due to the fact that Egypt during the 8th c. was pretty weak and useless. We can say that we know "for a fact" that Sennacherib led a martial campaign into the southern Levant toward the end of the 8th c. BCE. Not only does the biblical text record this account in Isaiah 36–39 (|| 2 Kings 18–20), but the Taylor Prism records the account from the Assyrian perspective and the palace reliefs from Nineveh depict the campaign sacking Lachish as well. This inscription tells us that Sennacherib locked Hezekiah up "like a bird in a cage." We know that the Omride dynasty was real thanks to things like the Black Obelisk of Shalmaneser III and the Mesha/Moabite inscriptions, which mention them directly. We also know—for a fact—that Assyria came in to northern Israel around 721 to deport the social elites and leaders, leading to an influx of northerners to the south shortly thereafter.

It would veritably take a full book to lay out the entire answer to your question (and indeed, such books exist!). Mazar's Archaeology of the Holy Land is a good place to start.