A lot of details have been given on this thread concerning the orthodox and traditional way of doing things, and I want to add some more insights regarding this.
A) As someone correctly pointed out, also within orthodox Judaism, the concept of having a big party and the bar mitzva boy reading from the Torah and/or being called up for maphtir (i.e saying the haphtara) is relatively recent - maybe a few hundred years. Also among some Eastern Sephardi communities (Aidot HaMizrach), a child from the age of nine or ten is already allowed to be called up for maphtir and say the haphtara, so this would not be a great deal with respect to Bar Mitzva which is age of 13, and the honors for a bar-mitzva boy in these communities is more towards reading from the Torah or leading the prayers.
B) With regards to the party, then among the very orthodox and now-a-days this is becoming more main-stream, it has become common place to make the main party on the Hebrew birthday date itself. Then the first call up, might be the next Monday or Thursday depending on the day of the week (or Rosh Hodesh) as has already been mentioned - and with this maybe a small celebration.
C) During the last Corona year with the uncertainties involved, I have seen a few bar-mitzva boys from "Rabbinical" families, make do with a simple call up on Shabbat so they should not spend hours learning to read the parsha from the Torah only to find the Health regulations, when the day arrives, not allowing to have a minyan (quorum of ten men).
D) On a personal note - I was brought up in a household that kind of defined itself as "semi-traditional". So, the issues that bothered my parents, in the early 1970s in London,UK, were:-
1) I was born on what is last day of Pesach in the diaspora. The issue was that the Rabbi's son was born a day or so after me, and my parent's realized that I would be completely out shadowed if I was called up on the next Shabbat.
2) It was expected that the main celebration would be on the Sunday following the Shabbat that I was being called up, since there were people coming from out of town and would be staying a few nights in London.
3) In England, it is the primary custom to keep the semi-mourning period of the Omer starting from the first day of Iyar, with a break on the semi-festival of lag ba'omer (33rd day of the Counting of the Omer). This would mean, that the main celebration (which to a certain extent had to conform to the religious issues) could not take place on this delayed date, as they wanted a band because that is what "everyone else" does.
4) So in the end, I was called up on the Shabbat preceding lag ba'omer and the main party was on lag ba'omer which was a Sunday that year.
Although, from my parents' point of view they wanted to give as much honor to the situation as they could, if I was in charge as a parent as I am now, I would have made the flow a whole lot simpler, regardless of with whom a bar-mitzva shabbat might be shared or how to schedule the party. I was at a bar-mitzva party once, that was during the mourning period of the Omer, and instead of a band, there were "Swingle Singers" who enhanced the occasion.
Ma'aleh Adumim, Israel