Susan asked, "How was God's Word preserved during Judah's exile?"
When Judah was taken captive, God’s Word remained with them.
One of the key verses about this topic is Ezra 7:1-6. King Cyrus, and later King Darius, allowed the Hebrew people to return to Jerusalem and begin rebuilding the temple. Among those in Babylon was the prophet Ezra. Here is how the Bible describes him:
Now after this, in the reign of Artaxerxes king of Persia, Ezra the son of Seraiah, son of Azariah, son of Hilkiah, son of Shallum, son of Zadok, son of Ahitub, son of Amariah, son of Azariah, son of Meraioth, son of Zerahiah, son of Uzzi, son of Bukki, son of Abishua, son of Phinehas, son of Eleazar, son of Aaron the chief priest—this Ezra went up from Babylonia. He was a scribe skilled in the Law of Moses that the LORD, the God of Israel, had given, and the king granted him all that he asked, for the hand of the LORD his God was on him. (Ezra 7:1–6 ESV)
Notice the last part of that description. Ezra "was a scribe skilled in the Law of Moses that the LORD, the God of Israel.” Even though he was living in captivity, he was an expert on God’s Law.
In addition, Jeremiah was ministering during this time. And, of course, at the beginning of the captivity, faithful men like Daniel were part of the captives.
So, God’s Law was preserved during that time because some of the faithful Jews took the scrolls with them into captivity and continued to study and learn and obey.