Why doesn't a convert dip in the mikveh and then wait eight days until he gets his circumcision? The gemara (Yevamot 97b) says ger shnitgayeir kekatan shenolad dami (A convert is like a newly-born baby). So - ostensibly - he should dip in the mikveh, and after eight days he should have his bris, right?
Actually the order for a convert is: first have circumcision, then wait for that to heal, then immerse in the mikvah. The immersion is what finalizes the conversion.
The law of "a convert upon conversion is like a newborn" is limited to certain laws, primarily that Torah law regards the convert as no longer related to their prior relatives.
I can only answer from personal experience. Since I had been circucised as a baby, when I converted I only needed a ceremonial bris -- a slight cut to draw some blood -- and then I went directly to the mikvah. It was explained to me that the bris, being a Biblical commandment, comes first. The mikvah is learned from the Oral Torah, and therefore comes after the bris.