If we agree that the Northern Tribe settled in Assyria at least for the time being then we have to consider what happened AFTER Judah was led away into captivity and RETURNED to Jerusalem. This is laid out in Ezra and Nehimiah. Please consider the following:
Ezr 6:21 The sons of Israel who returned from exile and all those who had separated themselves from the impurity of the nations of the land to join them, to seek the LORD God of Israel, ate the Passover.
Ezr 8:18 According to the good hand of our God upon us they brought us a man of insight of the sons of Mahli, the son of Levi, the son of Israel, namely Sherebiah, and his sons and brothers, 18 men;
Neh 1:6 let Your ear now be attentive and Your eyes open to hear the prayer of Your servant which I am praying before You now, day and night, on behalf of the sons of Israel Your servants, confessing the sins of the sons of Israel which we have sinned against You; I and my father's house have sinned.
Neh 2:10 When Sanballat the Horonite and Tobiah the Ammonite official heard about it, it was very displeasing to them that someone had come to seek the welfare of the sons of Israel.
Neh 8:17 The entire assembly of those who had returned from the captivity made booths and lived in them. The sons of Israel had indeed not done so from the days of Joshua the son of Nun to that day. And there was great rejoicing.
There are far many more verses about the "return" of the sons of Israel in Ezra and Nehimiah. This is NOT dogma. It is historical fact as laid out in the scriptures.
No doubt there were members of all tribes connected to Judah...and some of Judah who joined unto Israel. Just like in the civil war, some Confederates were northern sympathizers with the south and some southerners joined to the north.
Yet something else also happened after the return of the Babylonian exile. Israel had already been given their papers of divorce, now the brotherhood is being broken.
Zech 11:14 Then I cut asunder mine other staff, even Binders, that the brotherhood between Judah and Israel might be broken.
The house of Israel was settled in Assyria, not really captive but as land tenants. Many became very successful there and remained. Many eventually left and as Hoshea says Israel was swallowed up among the nations.
Yet even during the time of the Apostles, the house of Israel was still dispersed as James states in the intro to his epistle. Even Yeshua says "I am sent only for the lost sheep of the house of Israel." Why? They were those who had been divorced from covenant and must be redeemed by a kinsman. Judah was never divorced, not because they didn't err just as bad as Israel, but because the promise was that Messiah would come through the lineage of David.
Essentially, we know that this great ingathering has not taken place since the prophets describe it as a day that shall be exceeding great, that when Judah and Israel are united, never again shall they be driven from the land and that they shall all know YHWH. Blessings