Being a woman, I haven't ever been through it, but I do know that you have to have it done before your conversion (from what I've read, you do it before you have your mikvah, but it might be before or after the beit din).
It's not an issue for most American male converts, since most men in America are circumcised at birth (upwards of 95% of boys were circumcised in the 1950's-70's, then it's gradually declined to 65% of boys born now); all they have to do is give a drop a blood.