Milk and cheese are appearenlty two issues. (translations from Sefaria).
In Mishna AZ 2.6. there is a mention of Milk of Akum.
אלו דברים של גוים אסורין ואין איסורן איסור הנאה, חלב שחלבו גוי ואין ישראל רואהו, והפת, והשמן שלהן. רבי ובית דינו התירו בשמן.
Milk not watched is prohibited only for consumption.
The Chashash is a doubt perhaps milk of non kasher species is mixed.
But for cheese, AZ 2.4-5
המורייס וגבינת בית אונייקי של גויים--אסורין, ואיסורן איסור הניה, דברי רבי מאיר; וחכמים אומרין, אין איסורן איסור הניה.
The fish-brine and Bithynian cheese of non-Jews are prohibited and it is prohibited to derive any benefit from them, according to Rabbi Meir. The Sages say: Their prohibition is not against deriving benefit from them.
ב,ה אמר רבי יהודה, שאל רבי ישמעאל את רבי יהושוע כשהיו מהלכין בדרך, מפני מה אסרו את גבינת הגויים. אמר לו, מפני שמעמידין אותה בקיבת נבילה. אמר לו, והלוא קיבת העולה חמורה מקיבת הנבילה, ואמרו כל כוהן שדעתו יפה שורפה חיה; ולא הודו לו, אלא אמרו לא נהנין ולא מועלין. אמר לו, מפני שמעמידין אותה בקיבת עבודה זרה. אמר לו, אם כן, למה לא אסרוה בהניה. והשיאו לדבר אחר; אמר לו, ישמעאל אחי, היאך אתה קורא "כי טובים דודיך מיין" (שיר השירים א,ב [the proof is from the next words (Rashi) על כן עלמות אהבוך]), או כי טובים דודייך מיין. אמר לו, כי טובים דודייך מיין. אמר לו, אין הדבר כן, שהרי חברו מלמד עליו, "לריח שמניך טובים" (שיר השירים א,ג).
Rabbi Yehuda says: Rabbi Ishmael asked Rabbi Yehoshua [a question] as they were walking along the road. He said to him, "What is the cause for the prohibition against the cheese of non-Jews?" He said to him, "Because they curdle it inside the stomach of carrion." He said to him, "But is not [the law regarding] the stomach of a burnt offering more stringent than the stomach of carrion?! And they [the Sages] said: [It was proposed that] a priest with a good disposition may burn [a burnt offering after sucking out the fat from its stomach] while it it still raw [and has not yet been burnt with the offering, which would forbid one form deriving any benefit from it]. And they [the sages] did not agree with him [who proposed this opinion], but they said: One may not derive benefit [from the fats of the stomach], nor is one [who does so] liable for meilah [deriving prohibited benefit from from a sanctified object]." [Implying that one should therefore not be liable for the less stringent case of a stomach of carrion.] He [Rabbi Yehoshua] retracted, and [instead] said, "Because they curdle it in the stomachs of calves that were used for idolatry." And he [Rabbi Ishmael] said, "If so, why is there no prohibition to benefit from it?" He [Rabbi Yehoshua] redirected him to another topic. He said to him, "Ishmael, my brother, how do you read (Shir HaShirim 1): 'For dodechah [Heb. masc: "your love"] is better than wine,' or 'For dodayich [Heb. fem: "your love") is better than wine'? He said to him:, "'For dodayich is better than wine.'" He said to him, "The matter is not so. For its fellow [the following verse] teaches about it: 'For the fragrance of shemaneikha [Heb. masc: "your oils"] is good.'"
Bartenura says
שם כפר, שרוב עגלים הנמצאים שם קרבים לעבודה זרה. ר' מאיר חייש למיעוטא וסבר דאע"ג דרוב עגלים הוו מיעוטא כנגד שאר בהמות, חיישינן למיעוטא, וכל הגבינות הנמצאות שם אמרינן דלמא בקיבת עגלי ע"ז העמידן. וחכמים לא חיישי למיעוטא. ואין הלכה כר' מאיר בכל הני תלתא בבי:
In Mishna 4. There was a place in which a part of livestock, a minority were dedicated to Idolatry. There was a Chashash according to Rabbi Meir (who thinks that the din is safek rennet of idolatry despite the minority of idolatry). But for there is only an Isur to eat them.
What is the prohibition?
In Mishna 5, From the conclusion of the Gemara, explained in Rambam and Bartenura comments: The problem is "rennet from nevela", gastric tissue from a non ritually sacrified animal. This rennet is a part of the body. But the rennet mentioned by Rabbi Yehoshua isn't. This gastric content, it's not part of the body as excrement isn't. This explanation for the Chashash is not elucidated by the story reported in Mishna. Why? Gemara and Mefarshim explain. There was an official law decree for cheese from non Jews. To make the decret stronger, Rabbi Yishmael keep the secret about the true reason of the decret until its broadcast. After the broadcast of the decret, the reason was revealed. The Chashash is because of rennet made by gastric tissue, a part of the body of nevela. What is the principle of this prohibition, the rennet doesn't give taste in cheese? The explanation is for next questions.
We see from the sources that there was a separate problem for cheese, the rennet.
For Halacha. Some poskim allow milk from goym noweday, because the production is from kasher animals only. So, if a Jew make cheese with this milk they certainly allow it. But to buy from goym cheese is prohibited because of the decret.