There are two houses Israel AND Judah,who may I remind you at one time were at war with each other.
Now I wanna start this from the beginning
There are two tree's you and I belong to one
Both houses were present together before the Babylonian captivity. 2 Chronicles gives ample examples of many of the Northern tribes who wanted to remain faithful to the Lord migrated south and joined with the southern tribes. One of those migrations over 100 years after Sargon of Assyria captured the northern kingdom. So that is evidence that Sargon did not scatter the northern kingdom about.
When Ezra brought back his remnant from Babylon, he called them Jews 9 times and Israel 40 times.
When Nehemiah brought back his remnant from Babylon, he called them Jews 11 times and Israel 22 times.
Yeshua said that it was both houses that rejected Him to cause Him to return to His place. Likewise, it will be both houses combined that will be required to acknowledge that offense and petition for His return before He will...
Hosea 5:14-15 (NKJV) For I will be like a lion to
Ephraim,
And like a young lion to the house of
Judah.
I, even I, will tear them and go away;
I will take them away, and no one shall rescue.
15
I will return again to My place
Till they acknowledge their offense.
Then they will seek My face;
In their affliction they will earnestly seek Me."
"they" is both houses. So both houses were present in the land when Yeshua was here based on the above. 4 tribes are specifically referred to in the NT, one of them, Asher, is of the northern kingdom.
While we have been grafted in to the blessings of Israel thru the Messiah, that doesn't make us physical Israel or Jews that rejected Messiah and need to acknowledge that offense and petition for His return. Paul is pretty clear about that.
And the proof that those grafted into the blessings of Jacob/Israel are NOT Judah and Israel is shown in the Hosea passage above. The same people of Ephraim (Israel) and Judah that rejected Messiah will be the same people of Ephraim and Judah that must acknowledge that offense and seek Him. And in the "time of Jacob's trouble" in their affliction they will seek Him.
Matthew 23:37-39 (NKJV) "O Jerusalem, Jerusalem, the one who kills the prophets and stones those who are sent to her! How often I wanted to gather your children together, as a hen gathers her chicks under her wings, but you were not willing! 38 See! Your house is left to you desolate; 39 for
I say to you, you shall see Me no more till you say, 'Blessed is He who comes in the name of the Lord!'"
Just like the Lord has always kept a faithful remnant of Jacob/Israel, there are those of Jacob/Israel who have accepted Yeshua as the Messiah and are true Israel, but that doesn't make all who come to faith in Messiah "true Israel". And it doesn't negate that physical Jacob/Israel will still have to acknowledge its offense of rejecting Yeshua and petition for His return before He will. It is the primary reason that Satan has exerted so much effort to eradicate the Jewish people over the centuries. Unfortunately, the "church" has been a party to this at times. Mostly guided by theology that states the "church" has replaced Israel or is the "true Israel" and those who claim to be Jews and Israel are imposters.