Our Gemara on Amud Beis rules that conversion to Judaism requires a Bais Din of three Jewish judges. Conversion without this criterion is not valid.
The Gemara (Sotah 12b) tells us that when the daughter of Pharaoh went to the Nile to bathe, she also wanted to cleanse herself of her family's idolatrous practices. We can either understand this to be how she merited to be the adoptive mother of Moshe, or she intuited that a holy destiny awaited her, and she sought to purify herself prior. Interestingly, Rashi (ibid) is not satisfied with this narrative and adds, "She immersed herself with the intention of conversion."
The Sefer Daf al Daf brings down a question on this Rashi. How could she have performed a conversion on herself without a Bais Din? He quotes the Turei Even (Megillah 13a) who answers that conversion only requires a Beis Din to bring a Noachide into the Jewish nation. Since at that time, prior to the giving of the Torah and the covenant, even Jews were Noachides, no Bais Din was required.
However, this answer still begs the question as to what Rashi was trying to add. If it was purification she sought, the Gemara already stated that. If it was conversion, albeit not a Noachide to Jewish conversion, but a "Noachide son of Noah" to "Noachide son of Abraham conversion," what significance does this add beyond the Gemara? I believe that Rashi wants us to know that Pharaoh's daughter’s actions constituted something spiritually equivalent to conversion in every sense of the word. She either intuited via Ruach HaKodesh that immersion activated something beyond mere purification, or it was an ancient spiritual tradition stemming back to Adam, shared by many nations at that time, that immersion accomplished spiritual cleansing and possibly an act of conversion. (There is a theologically valid idea that Noach and the Patriarchs each had a separate "Torah" that they observed, distinct from the Torah of Moshe, see Sefer Hikkarim 1:4.)
If we accept this idea, that there was a Torah of sorts, transmitted from Adam or other early prophets such as Shem and Ever, we can resolve an age-old question on the Rambam. The Rambam famously asserts in the Moreh (III:46) that the rituals of animal sacrifice were a concession to the Jewish dependence and familiarity with worship via concrete actions, which were vestigial from idolatrous practices. The Rambam asks, imagine a person being told he can be religious without any observances that he is used to, such as Shabbos, fast days, or Tefilin. It would be utterly disorienting and unsustainable. So Hashem legislated a form of worship and devotion via means that were already familiar to the Jews, similar to the principle of the Torah speaking in common vernacular and idiom (Berachos 31b), so too the Torah used sociological and psychological touchstones to allow for a more seamless transition to monotheistic worship.
The Ramban (Vayikra 1:9) is horrified by such a position and finds it theologically absurd. The sacrificial rituals have deep mystical secrets and effects and must not be dismissed as mere leftovers from an idolatrous past. Furthermore, the Ramban asks what in yeshiva we would call a "Bomb Kashe" on the Rambam. What about the sacrifices brought by Cain, Hevel, Noach, and Avraham? Is it tenable to say at such an early time in history that they too were contaminated by idolatry?
How could the Rambam, with his towering intellect, have missed such an obvious question? He must have had a different way of understanding these early sacrifices. If we apply the above idea which emerges from Rashi in Sotah, we can explain the Rambam's line of thought as follows. There was a universal human religious instinct to offer sacrifices, analogous to the human religious instinct to purify oneself by bathing. At that point in history, it was not particularly Jewish. Therefore, for the Rambam, it was not a problem or repulsive that Adam and Noach and the Patriarchs utilized sacrifices because they were expressing a religious instinct and worshiping God, much like one might invent their own prayers or rituals to serve God in their own way. However, his argument still was that it was not a particularly Jewish feature, nor more significant than any man-made expression of devotion to God. If Cain decided to light a lamp in devotion to God, or Noach decided to make a rain dance (ironically!), that too would have been equally valid insofar as their devotion was sincere and enacted, but not more religiously powerful than anything else they might have done. Sacrifices only became specifically holy when mandated by the Torah. Thus, the Rambam's argument was that the only reason why it was mandated in such a particular physical way was that there was a pre-existing custom and religious impulse in that direction. True in the beginning, it was not idolatrous. Still, shortly enough, in the times of Enoch, it became idolatrous (see Rambam Laws of Idolatry 1:1). Thus God did not require or specifically need sacrifices as a mode of expression in ancient times prior to the giving of the Torah, and the Rambam can still argue that the Torah chose this mode of religious expression to ease the transition from concrete idolatrous worship of physical Gods to the monotheistic worship of an invisible, abstract God.
(There are, no doubt, other explanations for the Rambam's position. The Rambam in his introduction to the Moreh explicitly states that there are secrets to the Torah that he cannot divulge, and he will deliberately place contradictions within the text to hint at ideas that must be understood by wise inference only for those who are ready, and not by being stated outright. This is likely one of those examples.)
Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation 
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