First, an important correction: the text here (at least the Masoretic version) does not actually mention Judah; that appears to be an editorial addition in the NLT. The text of v14 is:
כֹּל, הַמִּשְׁפָּחוֹת הַנִּשְׁאָרוֹת--מִשְׁפָּחֹת מִשְׁפָּחֹת, לְבָד; וּנְשֵׁיהֶם, לְבָד.
JPS 1917: All the families that remain, every family apart, and their wives apart.
So verses 12-14 call out the families of: David, Nathan, Levi, and the Shimeites (and then "all the families that remain").
There are two possibilities for Nathan: the prophet, and David's son. If "Shimeites" refers to David's son Shammua, that might suggest that Nathan means David's son for parallelism. Further, Rashi suggests (below) that "all the families that remain" means all the other families of the house of David; if he is right, then that's a further argument for Nathan not being the prophet.
If we follow that theory, then the text could have just said the house of David and that would have included everybody. Prophetic writings sometimes use parallel formations for poetic effect even if they are not logically necessary, so that could be what is going on here. I do not know, however, why Nathan and Shammua would be particularly singled out among David's sons.
This leaves the question of why the Levites, who are not David's tribe, would be called out specially. I weakly speculate that they, being the overseers of the temple, have a special relationship to Jerusalem; just as King David and his family is named, perhaps they are too. But Zechariah can be a pretty hard book to understand, so I could well be wrong.
The medieval scholar Rashi (Rabbi Shlomo ben Yitzchak) offers the following comments on these verses:
the house of Nathan: the prophet. Some say that it refers to Nathan the son of David, as it is said (II Sam. 5:14): “Shammua and Shobab, and Nathan and Solomon.”
the house of Levi: The priests and the Levites.
the family of the Shimeites: Shammua the son of David. Scripture first makes a generality about the house of David, and then it specifies each one.
All the remaining families: of the house of David.
Please note: This answer was written for a neutral, academic audience
and is not intended to be interpreted in the context of a religious
belief or doctrine.