The Talmud (Kiddushin 29a) states the following:
למולו מנלן דכתיב וימל אברהם את יצחק בנו והיכא דלא מהליה אבוה מיחייבי בי דינא למימהליה דכתיב המול לכם כל זכר והיכא דלא מהליה בי דינא מיחייב איהו למימהל נפשיה דכתיב וערל זכר אשר לא ימול את בשר ערלתו ונכרתה
‘To circumcise him.’ How do we know it? — Because it is written: And Abraham circumcised his
son Isaac. And if his father did not circumcise him, Beth din is bound to circumcise him, for it is
written: Every male among you shall be circumcised. And if Beth din did not circumcise him, he is
bound to circumcise himself, for it is written: And the uncircumcised male who will not circumcise
the flesh of his foreskin, that soul shall be cut off. (Soncino translation)
This is reformulated by Rambam (Hilchot Milah 1:1-2) as follows:
ומצוה על האב למול את בנו ועל הרב למול את עבדיו יליד בית ומקנת כסף עבר האב או האדון ולא מל אותן ביטל מצות עשה ואינו חייב כרת שאין הכרת תלוי אלא בערל עצמו ובית דין מצווים למול אותו הבן או העבד בזמנו ולא יניחו ערל בישראל ולא בעבדיהן
A father is commanded to circumcise his son, and a master, his slaves. This applies both to those who are born in his home and to those purchased by him. If the father or the master transgressed and did not circumcise them, he negated the fulfillment of a positive commandment. He is not, however, punished by karet, for karet is incurred only by the uncircumcised person himself. The court is obligated to circumcise that son or slave at the proper time and should not leave an uncircumcised male among the Jewish people or their slaves. (Touger translation)
אין מלין בנו של אדם שלא מדעתו אלא אם כן עבר ונמנע למולו שבית דין מלין אותו בעל כרחו נתעלם מבית דין ולא מלו אותו כשיגדל הוא חייב למול את עצמו וכל יום ויום שיעבור עליו משיגדל ולא ימול את עצמו הרי הוא מבטל מצות עשה אבל אינו חייב כרת עד שימות והוא ערל במזיד
We may not circumcise a person's son without his knowledge, unless he has transgressed and did not circumcise him. [In such an instance,] the court must circumcise [the child] against [the father's] will.
If the matter does not become known to the court and they do not circumcise him, when [the child] reaches bar mitzvah, he is obligated to circumcise himself. With each and every day that passes after he has reached bar mitzvah, he negates a positive commandment. He is not, however, liable for karet until he dies uncircumcised, having intentionally [failed to perform the mitzvah]. (Touger translation)
This is echoed by the subsequent codes: Tur (Y.D. 261), Shulchan Aruch (Y.D. 261:1), Levush (Y.D. 261:1), Aruch Hashulchan (Y.D. 261:1).
It seems from these sources that there is a hierarchy wherein first the father is obligated, then the court if the father doesn't do it, and then the boy himself when he attains majority if the court doesn't do it. And, moreover, it appears that no one else is allowed to jump in and perform the circumcision instead of the father, unless he deliberately refuses to have his son circumcised, because as explained by the commentaries it is primarily his mitzvah so anyone else would be stealing his mitzvah (and would even be fined for doing so).
But this doesn't seem to directly address the case of an abandoned baby. Based on the above it might seem as if it is simply the court's responsibilty to take care of the circumcision, and if he is still uncircumcised when he attains majority then it becomes his own problem.
However, in the next seif in Aruch Hashulchan we find the following, which does seem to directly address your case:
דבר פשוט הוא שבמקום שאין ב"ד ואין אב החיוב מוטל על כל ישראל שבאותו מקום למולו דהא כתיב המול לכם כל זכר ולכל ישראל נאמרה אלא דכשיש ב"ד ממילא דהחיוב יותר על הב"ד
It is obvious that in a place where there is no court and no father, the obligation devolves to every Jew in that place to circumcise him, because the verse states "every male among you shall be circumcised" and this was said to all of Israel. It's just that if there is a court then obligation happens to be stronger on the court [than on an individual].
So in short, it would seem that a father can do the circumcision himself or appoint someone else to do it. No one can usurp the father and do the circumcision on their own unless the father is derelict in his duty. If there is a court around then it becomes the court's responsibility to appoint someone, but if there is no court than anyone has the right — and the obligation — to perform the circumcision.