There are plenty of sources for the idea that the celebration for the Bris should take place at the time of the Bris.
However, I have had the opportunity to preside over brisses where the parents were understandably quite nervous about the whole idea, wanting to give their son undivided attention at the time of the bris, not wanting to "spoil" this incredibly intimate family moment with the challenge of being hosts for a party.
The Bris, of course, was not delayed, as that is described in the Torah as being the 8th day of life (unless there are medical conditions which cause a delay). But the PARTY isn't Biblical... right? Or is it?
The source for having a celebration is the verse that Avraham made a big party on the day that הגמל את יצחק. Is that the bris? Is that a weaning? Is that a 'bar mitzvah'? (All of these are possibilities laid out in Midrashim and commentaries.)
Chasam Sofer writes this:
תורת משה בראשית פרק כא פסוק ח
ועתה כשהי' יצחק בן כ"ד ונגמל מאליו ברי אולם, אז עשה סעודת ברית מילה על יום הגמ"ל את יצחק בנו והיתה סעודת מילתו ביום כ"ד חדש
When Yitzchak weaned himself from his mother at 24 months and was clearly healthy and fully recovered from the Bris, that's when Avraham made the party for the Bris.
I don't know how many people would be interested in waiting 2 years - but this is certainly a novel idea, to wait until the child's second birthday to celebrate his bris!