If all Jews were descendants of the Tribe of Judah, then what precisely happened to the other tribes?

This is your first misconception. The term "Jew" as it is used in the modern day doesn't refer to descendants of the tribe of Judah. It refers to inhabitants of the Kingdom of Judah.

The united Kingdom of Israel only existed for a little more than a hundred years (from 1047-930 BCE), until it split into two separate kingdoms: the southern Kingdom of Judah (which included Jerusalem) and the northern Kingdom of Israel (which the Bible sometimes refers to as "Samaria," presumably to avoid confusion). The Kingdom of Judah was the smaller of the two, and included the territories of the tribes of Judah and Benjamin. There were Levites in the Kingdom of Judah too, but the tribe of Levy was more of a priestly caste than a tribe with territory. The later Kingdom of Israel (which I'm going to call "Samaria" so we don't get confused) included the territory of the other ten tribes (Reuben, Menasseh, Ephraim, etc.). Samaria was conquered by the Neo-Assyrian Empire around 722 BCE.

The Kingdom of Judah continued to exist until around 587 BCE when it was conquered by the Babylonian king Nebuchanezzar. The subsequent "Bablyonian Exile" was the period in which at least some of the inhabitants of the Kingdom of Judah were taken to Babylon as captives following Nebuchanezzars conquest (likely this was mainly the Jewish inhabitants of the city of Jerusalem).

Babylon was eventually conquered itself by the Persian emperor Cyrus the Great. According to the Biblical narrative, Cyrus then allowed the captive Jews and their descendants to return to Judah, provided they swore allegiance to him. There is some corroboration of an event like this taking place in the form of the Cyrus cylinder, which was a royal edict returning various captive peoples to their ancestral homelands (although the Jews are not mentioned).

Judah changed hands a lot over the next few centuries. It was ruled by the Persians until it was conquered by Alexander the Great, and then became part of the Selucid Empire, and eventually was conquered by the Romans. And eventually would become the birthplace of Jesus, and the place where Paul did a lot of his missionary work.

But to answer your question, the inhabitants of the Kingdom of Judah that were supposedly returned to their homeland by Cyrus the Great were from three tribes: Judah, Benjamin, and Levy. The Levitical priesthood was passed down patrilineally, and it was considered particularly important to maintain that bloodline. So as it turns out, you will actually find many Israelis (and Jews around the world) who will tell you that they are Levites (I believe Levites have been estimated to be ~5% of total Jews, but I can't find a source for that right now). But the vast majority of Jews are from the tribes of Benjmain and Judah.

By the end of the Second Temple period, and especially after the destruction of the Second Temple in 70 CE, the distinctions between members of the tribes of Benjamin and Judah largely ceased to be important. There probably are still Jews descended from Benjamin and Judah, but at this point nobody can tell who is who, and nobody really cares.

EDIT: I made an error in a (now deleted) comment to this post that I feel the need to correct. I said that the modern ethnic group of "Samaritans" was unrelated to the Kingdom of Samaria. That was incorrect; they likely are in some capacity (even if not necessarily genetically).

They claim descent from Ephraim and Menasseh, two of the tribes in the Northern Kingdom, and are referenced in 2 Chronicles 31:9 as being related.

I had misremembered something I learned years ago about the etymology of their name, which is the subject of some controversy. This is unrelated to their ethnic origins. I apologize to the person I wrongly corrected. This is a very stupid mistake and I'm sorry.

u/RepresentativePop has already given an excellent answer on the Jewish diaspora per se, but to add to it, there is another group who claim descent from not a Jewish tribe (not Judah, Benjamin or sometimes Levi) but one of the northern tribes: the Samaritans, less than a thousand of whom still exist in Israel and the West Bank. Y-DNA analysis confirms that they share a great deal of ancestry with Jewish lineages and seem to diverge from the parts of interest around the right time frame of the Assyrian conquest of the Northern Kingdom of Israel. Their matrilineal heritage (mtDNA haplotypes) are shared more with Middle Eastern groups, especially from Iraq, possibly because the Assyrians brought non-Israelite women in. They traditionally claim descent from the tribe of Ephraim, one of Joseph’s sons. They have their own, very slightly different, version of the Torah in a very slightly different ‘Samaritan Hebrew’, and hold Mount Gerizim in high regard over the Temple Mount or Mt Zion. Within the Abrahamic Religions it seems fair to either speak of a Judeo-Samaritanism or ‘Israelite’ religion, or to consider this the one Abrahamic religion that is a (now 20,000 times smaller) sister, rather than an historical offshoot (in some sense), of Judaism, with a similar historical time depth, but this may boil down to semantics.

Now there are other groups in Ethiopia and India who claim such descent, but genetic evidence hasn’t established any firm connection to the ancient Israelites yet, rather than with local populations, though this doesn’t preclude a small component that formed a cultural core. However, many major ethnic groups have had people make dubious claims of such descent like the British Israelites (for the British), the Black Hebrew Israelites (for West Africa), Germanic and other ‘white’ people in general (variations of Christian Identity)... even groups in Japan, and the Mormon and some conquistadors’ claims about Native American origins, have all used the lost ten tribes as a compelling and ‘available’ possible origin. Mixed with some of these have been other claims of ‘simply’ Jewish origin.

Another point to add is that genetic studies have also been conducted on Kohanim, or Cohens, who claim descent from Aaron (Moses’ brother, who, according to the Bible, was given the special privilege by God to be the ancestor of the Jewish ‘priestly caste’). Indeed it turns out that the majority they do form a specific Y-DNA subgroup with their own markers, the so-called Cohen Modal Haplotype, with an ancestor around 3000 years ago, about the expected ballpark... but with a few centuries’ uncertainty. Interestingly, the one remaining Samaritan Cohen family does not share this haplotype - but given the small sample size (this one lineage just needs one ‘illicit affair’ by a Cohen’s wife in a whole string of generations to be changed completely...), this doesn’t speak to Samaritan lineage overall.