For centuries the Jewish people were divided into two warring nations, Israel in the north, and Judah in the south. The best explanation for the existence of variant traditions in the Bible is that traditions evolved separately in each nation during this time, and were later edited together when the nations were once again joined as one.
Before I continue it might be a good idea to refute one possible objection that could be raised against the above described scenario. Were not the ten tribes of Israel ‘lost'. Presumably, if they had their own manuscripts they would have been lost with them. There are two possible points in time in which the melding together of Israeli and Judean manuscripts could have taken place. Immediately after the fall of Israel to the Assyrians, and the Israeli exile, a flood of refugees could have headed south into Judah, taking their traditions with them. This is a plausible explanation, and both Jeremiah and Ezekiel testify to the continuing presence of Israelites in Judah before the Babylonian invasion. (Although there is good evidence that the manuscripts were edited after the return from exile in Babylonian, this would have been a second redaction.)
The story of the ‘ten lost tribes of Israel' is a myth. The myth serves the purpose of tying currently living groups of people to Biblical prophecy (by insisting that their nation is one of the ten lost tribes, thus providing what is usually called a ‘secret key to unlock prophecy' as well as providing a personal ‘connection' between a living group and ancient prophets). For example prophecies referring to Ephraim can be said to be actually referring to Great Britain, for that is where this ‘lost tribe of Israel' migrated after the Assyrian invasion (it seems more likely that they would have gone south to Judah, and the manuscript evidence suggests that is what many of them did. Actually, any reference to Ephraim is a reference to the royal dynasty of Israel, since this is the tribe that produced Israeli monarchs and whose territory the capital was located.)
Ezekiel referred to both Israel and Judah being present in Judah and Jerusalem before the final Babylonian exile.
Then He said to me, "The iniquity of the house of Israel and Judah is very, very great, and the land is filled with blood and the city is full of perversion; for they say, ‘YAHWEH has forsaken the land, and YAHWEH does not see!' (Ezekiel Chapter 9 verse 9)
Jeremiah's prophecies directed toward the people of Israel show that rather than being off in England or some other land, the Israelites were still around during the time just before and even after the Babylonian captivity. According to the following prophecy, both Israel and Judah were in such oppressive bondage, that neither one was ever allowed to leave (so it would be difficult for Ephraim, for example, to become 'lost' and wander off to England.
"Thus says YAHWEH of hosts, "The sons of Israel are oppressed, And the sons of Judah as well; And all who took them captive have held them fast, They have refused to let them go. Their Redeemer is strong, YAHWEH of hosts is His name; He will vigorously plead their case So that He may bring rest to the earth, But turmoil to the inhabitants of Babylon. A sword against the Chaldeans," declares YAHWEH, "And against the inhabitants of Babylon And against her officials and her wise men!" (Jeremiah Chapter 50 verse 33)
In the following prophecy Israel is invited to return to the land.
"And YAHWEH said to me, "Faithless Israel has proved herself more righteous than treacherous Judah. Go and proclaim these words toward the north and say, ‘Return, faithless Israel,' declares YAHWEH; ‘I will not look upon you in anger. For I am gracious,' declares YAHWEH; ‘I will not be angry forever." (Jeremiah Chapter 3 verse 11)
In the following prophecy, Jeremiah describes how Israel was first crushed by Assyria, and then the Babylonians 'broke Israel's bones'. The only way Israel could first be crushed by Assyria and then crushed by the Babylonians (obviously at the same time that the southern kingdom of Judah fell to Babylon) is if the Israelites were in Judah, as both Jeremiah and Ezekiel testify that they were.
"Israel is a scattered flock, the lions have driven them away. The first one who devoured him was the king of Assyria, and this last one who has broken his bones is Nebuchadnezzar king of Babylon. Therefore thus says YAHWEH of hosts, the God of Israel: ‘Behold, I am going to punish the king of Babylon and his land, just as I punished the king of Assyria. And I will bring Israel back to his pasture and he will graze on Carmel and Bashan, and his desire will be satisfied in the hill country of Ephraim and Gilead. In those days and at that time,' declares YAHWEH, ‘search will be made for the iniquity of Israel, but there will be none; and for the sins of Judah, but they will not be found; for I will pardon those whom I leave as a remnant.'" (Jeremiah Chapter 50 verse 17)
This point is recapitulated in the following prophecy, which describes both Israel and Judah returning from Babylonian exile.
"The word which YAHWEH spoke concerning Babylon, the land of the Chaldeans, through Jeremiah the prophet: Declare and proclaim among the nations. Proclaim it and lift up a standard. Do not conceal it but say, ‘Babylon has been captured, Bel has been put to shame, Marduk has been shattered; Her images have been put to shame, her idols have been shattered.' For a nation has come up against her out of the north; it will make her land an object of horror, and there will be no inhabitant in it. Both man and beast have wandered off, they have gone away! In those days and at that time, declares YAHWEH, the sons of Israel will come, both they and the sons of Judah as well; they will go along weeping as they go, and it will be YAHWEH their God they will seek. They will ask for the way to Zion, turning their faces in its direction; they will come that they may join themselves to YAHWEH in an everlasting covenant that will not be forgotten." (Jeremiah Chapter 50 verse 1)
According to Ezekiel, Judah was guilty of ‘40 years of sin'.
"... bear the iniquity of the house of Judah; I have assigned it to you for forty days, a day for each year." (Ezekiel Chapter 4 verse 6)
Ezekiel testified that not only were Israelites to be found in Judah, they were also serving in the temple prior to the Babylonian destruction of Jerusalem.
"And He said to me, Go in and see the wicked abominations that they are committing here (note: in the temple). So I entered and looked, and behold, every form of creeping things and beasts and detestable things, with all the idols of the house of Israel, were carved on the wall all around. Standing in front of them were seventy elders of the house of Israel, with Jaazaniah the son of Shaphan standing among them, each man with his censer in his hand and the fragrance of the cloud of incense rising. Then He said to me, "Son of man, do you see what the elders of the house of Israel are committing in the dark, each man in the room of his carved images? For they say, ‘YAHWEH does not see us; YAHWEH has forsaken the land.'" And He said to me, "Yet you will see still greater abominations which they are committing." (Ezekiel Chapter 8 verse 9)
The Israelite exile happened in the century previous to the Babylonian exile of Judah. According to Ezekiel, Judah sinned for 40 years prior to this exile. This testimony is in general agreement with the testimony of Jeremiah, who suggested that it was Israel who ‘made Judah sin' by putting ‘detestable things' into the temple. Note that the two nations were divided for centuries and the Israelites had no part in the temple in Jerusalem, worshiping instead at Bethel. It is obvious that after Israel fell Israelites went south to Judah, and it is mentioned in the following passage that once again they were found in Jerusalem and in the temple. The only way that Israelites could have placed their abominations in the temple or have ‘caused Judah to sin' would be if Israelite refugees had migrated south after the Assyrian invasion.
"Indeed this city has been to Me a provocation of My anger and My wrath from the day that they built it, even to this day, so that it should be removed from before My face, because of all the evil of the sons of Israel and the sons of Judah which they have done to provoke Me to anger—they, their kings, their leaders, their priests, their prophets, the men of Judah and the inhabitants of Jerusalem. They have turned their back to Me and not their face; though I taught them, teaching again and again, they would not listen and receive instruction. But they put their detestable things in the house which is called by My name, to defile it. They built the high places of Baal that are in the valley of Ben-hinnom to cause their sons and their daughters to pass through the fire to Molech, which I had not commanded them nor had it entered My mind that they should do this abomination, to cause Judah to sin." (Jeremiah Chapter 32 verse 31)
There is also testimony to the presence of the Israelites in Jerusalem after the return from exile.
"Now these are the heads of the provinces who lived in Jerusalem, but in the cities of Judah each lived on his own property in their cities—the Israelites, the priests, the Levites, the temple servants and the descendants of Solomon's servants. Some of the sons of Judah and some of the sons of Benjamin lived in Jerusalem." (Nehemiah Chapter 11 verse 3)
According to the book of Ezra the people offered "a sin offering for all Israel 12 male goats, corresponding to the number of the tribes of Israel." (Ezra 6:17) He also describes how "The sons of Israel who returned from exile...ate the Passover." (Ezra 6:21) There are also lists of "the number of the men of the people of Israel" who returned from exile. (Ezra 2:2, Nehemiah 7:7 ) We are told that "The exiles who had come from the captivity offered burnt offerings to the God of Israel: 12 bulls for all Israel." (Ezra 8:35) Ezra 10:25 lists the names of the sons of Israel who married foreign women. "Of Israel, of the sons of Parosh there were Ramiah, Izziah, Malchijah, Mijamin, Eleazar, Malchijah and Benaiah ... " And as Nehemiah states, "And the rest of Israel, and of the priests and the Levites, were in all the towns of Judah." (Nehemiah 11:20)
Isaiah chapters 40 onward are a composition written by a prophet in the school of Isaiah at about the time of the return from Babylonian exile (prophecies are addressed to Cyrus in person, and the prophecy describes the return from exile). Certain fundamentalists like to dispute the dating of these chapters, by arguing that Isaiah was a prophet, and thus addressed Cyrus many centuries before he was born, but in any case, these prophecies describe the return of both Israel and Judah from exile, which, when you consider the above material, and when you consider how unlikely it would be for Judah to return and Israel to remain in captivity, is about what you would expect. (So to recapitulate, there are no ‘ten lost tribes of Israel'. This was a religious myth invented in the 18th century to give people a feeling of personal connection with prophecy by inventing ways to connect their own country with prophecy, by insisting they were mentioned there, but in ‘secret code'.)