This understanding of the verses in Hoshea loses sight of the framework of the first chapters of Hoshea. As the Talmud in Pesachim 87 put it (emphasis mine):
"The Holy One, Blessed be He, said to Hosea: Your sons, the Jewish people, have sinned. Hosea should have said to God in response: But they are Your sons; they are the sons of Your beloved ones, the sons of Abraham, Isaac, and Jacob. Extend Your mercy over them. Not only did he fail to say that, but instead he said before Him: Master of the Universe, the entire world is Yours; since Israel has sinned, exchange them for another nation. The Holy One, Blessed be He, said: What shall I do to this Elder who does not know how to defend Israel? I will say to him: Go and take a prostitute and bear for yourself children of prostitution. And after that I will say to him: Send her away from before you. If he is able to send her away, I will also send away the Jewish people. This deliberation provides the background of the opening prophecy in Hosea, as it is stated: “The Lord said to Hosea: Go, take for yourself a wife of prostitution and children of prostitution”...After two sons and one daughter had been born to him, the Holy One, Blessed be He, said to Hosea: Shouldn’t you have learned from the example of your master Moses, who, once I spoke with him, separated from his wife? You too, separate yourself from your wife. He said to Him: Master of the Universe, I have sons from her and I am unable to dismiss her or to divorce her. In response to Hosea’s show of loyalty to his family, the Holy One, Blessed be He, rebuked him and said to him: Just as you, whose wife is a prostitute and your children from her are children of prostitution, and you do not even know if they are yours or if they are children of other men, despite this, you are still attached to them and will not forsake them, so too, I am still attached to the Jewish people, who are My sons, the sons of My faithful who withstood ordeals, the sons of Abraham, Isaac, and Jacob. They are so special that they are one of the four acquisitions that I acquired in My world."
We find, therefore, that Hashem came to explain to Hoshea why, despite being angry at Yisrael's betrayal, He still keeps them as His nation. That's why in the end of Hoshea 2, the names are reversed:
"And the earth shall respond With new grain and wine and oil, And they shall respond to Jezreel. I will sow her in the land as My own; And take Lo-ruhamah back in favor; And I will say to Lo-ammi, “You are My people,” [Ami atah in Hebrew] And he will respond, “[You are] my God.”" (2:24-25)
As I wrote in the comments, Yechezkel uses the terms "Yisrael" and "Yehudah" almost interchangeably. That's why we find that when the elders of the community came to sit before him, sometimes they are called "elders of Yisrael" (for example, 20:1; 14:1) and sometimes "elders of Yehudah" (8:1). And when he sees the elders of those that remained in Judea in the time of Tzidkiyahu, he refers to them as "the elders of the House of Yisrael" (for example, 8:11). And in the Vision of the Dry Bones, he then sees them gaining new life - so it can't be that they were really cut off completely.
With that said, there are some sources that bear witness to that fact that some of the descendants of the Ten Tribes were still around during the Second Temple period:
Rabbi Prof. Shmuel Klein points out in Eretz Hagalil (Land of the Galilee), pg. 2, that there's evidence in Tanach that not the whole of the Kingdom of Yisrael was exiled by the Assyrians. We find that King Chizkiyahu invited them over for the Pesach celebration in Yerushalayim (Chronicles 2:30:1-11), as well as two later queens being from Zevulunite towns: Meshulemet bat Charutz of Yotva, wife of Menashe, and Zevudah bat Pedayah of Rumah, one of the wives of Yoshiyahu.
Tosfot in Gittin 36a state that as Yirmiyahu returned at least some of the Ten Tribes, and Yoshiyahu ruled over them, it means that some were around during later periods.
According to the Talmud Yerushalmi Taanit 20a and Beresheet Rabbah 98:8, a man named "ben Kovisin" (or a family called "Beit Kovshim" according to version in the midrash) could trace his lineage to King Achav; the descendants of King Yehu lived in Tzippori; and Rabbi Yossi ben Chalafta traced his lineage to Yonadav ben Rechav (though there's a disagreement as to what was Yonadav's own lineage, it seems apparent in Kings 2:10:15 that he and his family lived in the Kingdom of Yisrael, and later were some of those that moved to the Kingdom of Yehudah (per Yirmiyahu 35:1-6).
There's a discrepancy between the total number of returnees in Ezra 2:64 and the individual numbers of each household. Seder Olam Rabbah puts that down to the "extras" being of other tribes, and indeed, Ezra 6:17 states that sacrifices were made for all of the tribes of Yisrael and Chronicles 1:9:1-3 states that some of the returnees to Yerushalayim were people from Efraim and Menashe (and see Malbim on the chapter). As to why they weren't directly mentioned in Ezra 2, see here for one possible answer.
According to the Talmud in Pesachim 4a, some descendants of the Ten Tribes were identifiable even in Talmudic times according to the way they talked.
According to the Letter of Aristeas, the 72 sages that created the Septuagint consisted of six men from each of the twelve tribes.
But, there doesn't appear to be evidence that any of these people worshiped Hashem in any manner different from the majority of Am Yisrael.1 In fact, Rabbi Yossi ben Chalafta was a great sage, as was his father Rabbi Chalafta and some of his descendants as well, and of course, if what the author of the Letter of Aristeas wrote is accurate, the 72 sages were chosen because they were wise and God-fearing.
Likely some assimilated and became Hellenistic Jews, much like other Jews became Hellenistic Jews, while others remained regular Jews, like other regular Jews.
1 I did mention in the comments a theory by Rabbi Dr. Yehoshua Brand from his book "Glassware in the Talmud" (the second section of the book is dedicated to other subjects). The theory in short is that since it says in Chronicles 1:9:3 that people of Efraim and Menashe came to live in Yerushalayim after Cyrus's proclamation, and as there's a discrepancy between the total number of returnees and the individual numbers of each household that returned, it seems there's evidence that people from the Ten Tribes really did return, but were rejected by the people from the Kingdom of Yehudah. He ties to this certain prophecies in Zechariah that, according to him, can only be explained as referring to people from other tribes. The reason the people were rejected by the Judeans was because: a. The Judeans were proud and looked down upon these people. b. These people, having lived separate from the Judeans for about 150 years, had their Judaism develop in a different direction than that of the Judeans. This led to harsh arguments about how to worship Hashem in Yerushalayim. Tempers flared, and eventually Zechariah decided to kick them out, as these people weren't willing to submit to the Judean halacha. They were told to go live in their ancestral homes in northern Yisrael. Samaria was taken by the Samaritans (Cuthites and other nations), and so they headed further north to the Galiliee, which was part of the Aram-Damascus Persian province. They settled there and vowed revenge against the Judeans that they believed had wronged them, thus forming "The Damascus Covenant". The early layers of the text of the covenant are practically biblical in style - because these people were from the last years of the Tanach.
(I personally think this theory is way too wild. For one, it paints the picture of Zechariah basically being a big bully. There are other reasons, too, but I won't get into that).