Firstly, Was blood atonement relevant to the point of the book? Not really. Does it refute what Christians say about blood atonement? No.
"Jonah Goes to Nineveh
3 Then the word of the Lord came to Jonah the second time, saying, 2 “Arise, go to Nineveh, that great city, and call out against it the message that I tell you.” 3 So Jonah arose and went to Nineveh, according to the word of the Lord. Now Nineveh was an exceedingly great city,[a] three days' journey in breadth.[b] 4 Jonah began to go into the city, going a day's journey. And he called out, “Yet forty days, and Nineveh shall be overthrown!” 5 And the people of Nineveh believed God. They called for a fast and put on sackcloth, from the greatest of them to the least of them.
The People of Nineveh Repent
6 The word reached[c] the king of Nineveh, and he arose from his throne, removed his robe, covered himself with sackcloth, and sat in ashes. 7 And he issued a proclamation and published through Nineveh, “By the decree of the king and his nobles: Let neither man nor beast, herd nor flock, taste anything. Let them not feed or drink water, 8 but let man and beast be covered with sackcloth, and let them call out mightily to God. Let everyone turn from his evil way and from the violence that is in his hands. 9 Who knows? God may turn and relent and turn from his fierce anger, so that we may not perish.”
10 When God saw what they did, how they turned from their evil way, God relented of the disaster that he had said he would do to them, and he did not do it."
Are we told whether sacrifices were made by the Ninevites, No. Could they have done, yes and no.
They wouldn't of had the means or proper of atonement nor is there an indicator that they turned to the God of Israel, though what a fascinating idea that they would follow Jonah back to Israel after repentance toward God. We simply don't know
But other than that, Michael L Brown mentioned a rabbinic tradition in his dialogue with Shmuley Boteach on It's Supernatural and in his 2nd Volume of Answering Jewish Objections to Jesus (Which I have yet to read, I am currently on the first one at the time of this writing), He mentions Sukkah 55b where Yohannan mentions the destruction of the atonement and criticises the Romans for destroying it, which would indicate the temple was designed to make atonement for the other nations, at least according to the Rabbi in the tradition:
"א"ר יוחנן אוי להם לעובדי כוכבים שאבדו ואין יודעין מה שאבדו בזמן שבהמ"ק קיים מזבח מכפר עליהן ועכשיו מי מכפר עליהן:
Rabbi Yoḥanan said: Woe unto the nations of the world that lost something and do not know what they lost. When the Temple is standing, the seventy bulls sacrificed on the altar during the festival of Sukkot atones for them. And now that the Temple is destroyed, who atones for them?": https://www.sefaria.org/Sukkah.55b.11/en/William_Davidson_Edition_-_English?lang=bi
I leave you to judge the tradition for yourself.
Assuming Brown's interpretation of the tradition is correct, it does safely lay to rest the point that atonement in blood was made just not how we expect, namely Israel doing the atonement for the nations (assuming repentance is present within those nations.