This is a great question. In fact, you have asked four questions and I will try to give some thoughts on each one.
- Where Are the Lost Tribes?
- Are they spread around the nations?
- If they are spread around the nations, how are they Jewish/have they assimilated?
- Are they spread within the Jewish nation?
In answer to question 1: Where Are the Lost Tribes?
Below, I will give a summary of Biblical and Talmudic/Midrashic sources as to their whereabouts.
The Tanach places them in the following places as per the verse in II Kings 17:6
בִּשְׁנַת הַתְּשִׁיעִית לְהוֹשֵׁעַ לָכַד מֶלֶךְ־אַשּׁוּר אֶת־שֹׁמְרוֹן
וַיֶּגֶל אֶת־יִשְׂרָאֵל אַשּׁוּרָה וַיֹּשֶׁב אֹתָם בַּחְלַח וּבְחָבוֹר
נְהַר גּוֹזָן וְעָרֵי מָדָי׃
In the ninth year of Hoshea, the king of Assyria captured Samaria. He
deported the Israelites to Assyria and settled them in Halah, and in
Habor, at the River Gozan, and in the towns of Media.
Later in II Kings 18:11 it repeats these places:
וַיֶּגֶל מֶלֶךְ־אַשּׁוּר אֶת־יִשְׂרָאֵל אַשּׁוּרָה וַיַּנְחֵם בַּחְלַח
וּבְחָבוֹר נְהַר גּוֹזָן וְעָרֵי מָדָי׃
And the king of Assyria deported the Israelites to Assyria. He settled
them in Halah, and in Habor, the River Gozan, and in the towns of
Media.
When this is later recounted in I Chronicles 5:26, we see one of the previous locations removed and one new one added:
וַיָּעַר אֱלֹהֵי יִשְׂרָאֵל אֶת־רוּחַ פּוּל מֶלֶךְ־אַשּׁוּר
וְאֶת־רוּחַ תִּלְּגַת פִּלְנֶסֶר מֶלֶךְ אַשּׁוּר וַיַּגְלֵם
לָראוּבֵנִי וְלַגָּדִי וְלַחֲצִי שֵׁבֶט מְנַשֶּׁה וַיְבִיאֵם לַחְלַח
וְחָבוֹר וְהָרָא וּנְהַר גּוֹזָן עַד הַיּוֹם הַזֶּה׃
And the God of Israel roused the spirit of Pul, King of Assyria, and
the spirit of Tillegath-Pilneser, King of Assyria, and he deported the
Reubenites, the Gadites, and the half-tribe of Manasseh, and brought
them to Halah, and Habor, and Hara, and the River Gozan, to this day.
So from the Biblical sources themselves we have five places where they were deported, namely: Halah, Habor, the River Gozon, the towns of Media, and Hara.
The Talmud Sanhedrin 94a asks where were they exiled to:
להיכא אגלי להו מר זוטרא אמר לאפריקי ורבי חנינא אמר להרי סלוג
Where were they [the Ten Lost Tribes] exiled to? Mar Zutra says, “To
Afriki” and Rabbi Hanina says, “To the Mountains of Selug”.
It is worth noting here that Maharal (Netzach Yisrael 34:5) says the Afriki mention in the Gemara here is not the Africa we are familiar with but rather another place as per the Talmud in Tamid 32a which refers to a place we cannot reach.
In the Talmud Yevamot 16b-17a Rabbi Abba bar Kahana identifies the places mentioned in the Book of Kings (quoted above). He suggests Halah is Hilazon, Habor is Hadiyeb, the River Gozan is Ginzak, and the towns of Media is Hamadan and its surrounding areas. Some say it was Nihar and its surrounding areas. The Gemara asks which are these surrounding areas and Shmuel says they are Kerach, Mushkei, Hidkei and Domkei.
The Midrash (Pesikta Rabbati 31) says they were exiled to three places:
מהו לאמר לאסורים צאו ששלש גליות נעשו עשרת השבטים אחת גלתה לסמבטיון
ואחת גלתה לפנים מסמבטיון ואחת גלתה לדפנו של רבלתה ונבלעה שם
What does [the verse in Isaiah 49:9] mean “Say to the prisoners, ‘Go
free’”? [It teaches us] that there were three exiles [places] that
happened to the Ten Tribes. One was exiled to the Sambatyon, one was
exiled to beyond the Sambatyon, and one was exiled to Daphne of
Antiochia, and they were swallowed [assimilated] there.
A similar version of this is found in the Jerusalem Talmud (Sanhedrin 10:5):
לְשָׁלֹש גָּלִיּוֹת גָּלוּ יִשְׂרָאֵל. אַחַת לִפְנִים מִנְּהַר
סַנְבַּטְיוֹן וְאֶחָד לְדַפְנֵי שֶׁלְּ אַנְטִוֹכְיָא וְאַחַת שֶׁיָּרַד
עֲלֵיהֶם הֶעָנָן וְכִסָּה אוֹתָם.
There were three exiles for Israel. One was behind the Sambatyon
River, and one to Daphne of Antiochia, and one, the cloud came down
upon them and covered them.
This gives us a good idea in terms of the Biblical and Talmudic/Midrashic sources of where they were exiled to. It does not help us in terms of where are those places located and what happened to them there. Did they assimilate? Did they relocate afterwards? Did they maintain some vestiges of Jewish practice? This topic, however, has thousands upon thousands of books written on this subject and there are many theories which attempt to answer the question.
In answer to question 2: Are they spread around the nations?
Although we can not know definitively, it is highly plausible to suggest that over the more than two and a half millennia since their exile, there was likely a huge amount of movement around the world.
Dr Avigdor Shachan, in his book, In the Footsteps of the Ten Lost Tribes, (page 45) suggests there was a lot of population movement by the ten tribes after their initial exile by the Assyrians. He says, their first base was Afghanistan, directly east of Israel, where according to some, four of the tribes of Israel established an independent state. This was likely in the region of the Hindu-Kush mountains and the surrounding area. According to Dr Shachan they wanted to reach Japan, and calculated two distinct routes of travel from Afghanistan. The shorter northern route would take them eastward from Afghanistan, by passing north of the Himalayas and traveling through the Pamir mountains via Chinese Turkestan and the Gobi Desert, toward the Korean Peninsula, which was a scheduled meeting point for later expeditions. The southern route would bypass the Himalayas on the south, via what is now Burma (Myanmar), toward the eastern coast of the continent and moving north towards the Korean Peninsula. This southern route was longer than the northern route but much easier to traverse.
There are, of course, many academics (perhaps most) who suggest that the tribes are lost because they simply no longer exist; only their legend lives on in the hearts of those who seek them. But as an actual body of people they were assimilated shortly after the Assyrian deportations, as was the policy of the Empire to relocate and assimilate peoples in order to expand their empire.
There are numerous groups around the world who claim descent from the Ten Lost Tribes or others have ascribed this identity to them. Each will have some oral history or tradition as to how they arrived to be where they are. Some of these tribal histories are shrouded in mystery and difficult to prove beyond the oral history of the people telling them.
In answer to your question, if we assume that the Ten Lost Tribes did not all assimilate entirely, each group would need careful study to determine the validity of their claims and it is fair to say that with the amount of time that has passed, it is unlikely to reach a definitive conclusion.
Where are they? As one professor of history put it, “Political claims regarding the ten tribes status of various groups have been made since the early modern period, attaching themselves to real and imagined peoples from, literally, A-Z” (The Ten Lost Tribes, A World History, Zvi Ben-Dor Benite, pages 5-6).
Chronicling all of the groups would be a huge undertaking but to give a sense of just how far this goes, here are some of the more popular claims. From the Pashtun people of Afghanistan and Pakistan, the Bnei Menashe of Myanmar, to the Lemba in Zimbabwe and South Africa, to the Bene Israel of the western coast of India, to the Native Americans, the Beta Israel of Ethiopia, to the Igbo of Nigeria, to the Shinto of Japan, the Kyrgyz people, the Jews of Georgia and Bukhara; even the British people have been identified as lost tribes and adherents to what is known as Anglo Israelism still exist today.
In answer to question 3: If they are spread around the nations, how are they Jewish/have they assimilated?
Regarding whether they assimilated or not, consider the Talmud Yevamot 16b, which states:
אָמַר רַב יְהוּדָה אָמַר רַב אַסִּי: גּוֹי שֶׁקִּידֵּשׁ בִּזְמַן
הַזֶּה — חוֹשְׁשִׁין לְקִדּוּשִׁין, שֶׁמָּא מֵעֲשֶׂרֶת הַשְּׁבָטִים
הוּא
Rabbi Yehuda said in the name of Rav Asi: With regard to a gentile who
betrothed a Jewish woman nowadays, we are concerned that the betrothal
might be valid, despite the fact that a betrothal of a gentile is
meaningless, lest he be from the ten tribes of Israel who intermingled
with the gentiles.
We can see here that even in the time of these Talmudic discussions, which took place approximately 1,000 years after the exile of the Ten Tribes by the Assyrian empire, there was a clear understanding that assimilation had taken place. So much so that Rav Asi believes that someone who marries a gentile [in certain places where the exiles were sent to…see the Gemara further on] may have actually performed a halachic Jewish marriage as they may have assimilated among the nations but still be Jewish.
Another important point to consider is when discussing the Ten Lost Tribes, we are not necessarily talking about ‘halachic Jews’ or what we would consider to be a ‘full Jew’ according to normative Rabbinic Judaism today.
It is worth remembering that tribal heritage goes through the father, not the mother. Perhaps a productive way to consider it is that, they don’t have a [normative] Jewish identity - rather they have an Israelite identity, which is something you can have without being Jewish. An interesting fact to note, the Northern Kingdom [from which the Ten Tribes come] was called the Kingdom of Israel - and many of the groups who lay claim to be descendent have the name Israel (as opposed to Jew/Yehud) in their name. Like the Bene Israel of India, or the Pashtun of Afghanistan (they refer to themselves as Bani Israil), or ones who specifically trace themselves to a specific tribe.
The Beta Israel (Ethiopian Jews) maintained a very strong identity even though they were ostensibly cut off from mainstream Judaism for well over a thousand years. They had probably the strongest claim from any other group as they had actual texts (not everything) in their possession for thousands of years, including some form of an Oral Torah tradition. Rav Ovadia Yosef has argued they are descendent from the Tribe of Dan and much has been written about this which I will not bring here.
In any event, the identity of groups who claim to be Lost Tribes is a complex one, and one that is difficult to define in the way the question posed here attempts to.
In answer to question 4: Are they spread within the Jewish nation?
Most people within the Jewish nation today would be said to belong to either the tribes of Yehuda, Binyamin or Levi.
Some suggest that prior to the Assyrian conquest of the Northern Kingdom, many people from the Ten Tribes fled south to their brethren in the Kingdom of Judah and were mixed up among the tribes of Judah and Benjamin. See Otzar Midrashim, Midrashim on the Ten Lost Tribes, Introduction 2.
In addition, the Talmud Megilla 14b and Arakhin 33a describes how the prophet Jeremiah went to return many of the lost tribes in his day who were then integrated into the tribes of Yehuda and Binyamin and didn’t retain a separate tribal identity. In that case too, it is clear from the commentaries that it was only a minority of these tribes who returned, and their proof is that otherwise the mitzvah of Yovel (Jubilee) would have been reinstituted (as it requires the majority of the tribes living in their respective portions of the Land).
According to this opinion, the Ten Tribes are integrated within the Jewish people as we know them today.