As Shalom alluded to there are two ways to evaluate circumcision vis-a-vis a Ger. One is that it is part of the process of becoming Jewish (for a man). In that case, there is really nothing to start with - if he isn't getting a circumcision it isn't doing the process, he doesn't become Jewish.
However, in the case where he can't have a circumcision, there are two possible cases. One is that he lacks the anatomy (there could have been a previous accident, etc.). In that case he is viewed as it is with a woman - it doesn't apply to him. However, if he has the technical capacity to have circumcision, but could not with permission of the Torah - i.e. the procedure would kill him, or there is a substantial concern of that outcome - then he simply cannot convert. It is not that he can become Jewish and skip the circumcision. He can't fulfill the circumcision according to the Torah, so he can't become Jewish according to the Torah. (Or perhaps he can choose to risk his life according to the Torah in this case).
The second way of evaluating it is that the requirement of circumcision is a technical point - he is becoming Jewish, so he is required to be circumcised. He takes care of that first so he doesn't spend any time as a Jew without fulfilling that obligation. In that case, if the procedure has a substantial concern of killing him, the Torah says that he is exempt from this commandment, at which point the conversion can proceed without it. (There is another way of looking at this - if the Torah says he is exempt he can convert, just like he can convert without bringing the sacrifice since we don't have the Temple today and he is thus exempt from the sacrifice but can still convert - but in that case conversion without actually being exempt from circumcision is a non-starter just like the first evaluation).
In the case of someone who simply doesn't want to have a circumcision, according to the first evaluation above - there is nothing to talk about. According to the second evaluation, it would get into a different discussion. One of the fundamental requirements of conversion is accepting the obligations of the Torah. If someone is rejecting any of it, the conversion is not valid. So if someone doesn't agree to be circumcised, he is rejecting a Mitzvah and would thus invalidate any conversion ceremonies. Note that it isn't enough to agree that in theory G-d commanded them. The convert actually has to intend to do them. Here the refusal is clearly showing that he has no such intention.
There is an edge case to discuss here - he converts not knowing that he has to be circumcised. Because while the convert has to accept upon himself to do all of the commandments, he doesn't have to actually know what they all are. According to the second evaluation, he might well end up being Jewish. That would be a shame, because while as a non-Jew he could have had a share in the world to come, as an uncircumcised Jew (at least as the Rambam defines Kares) he would not.
As for which commandment - there is a constant obligation for a Jewish male to be circumcised from 8 days on (Vayikra 12:3). See what I wrote about that here.