It mentions this subject on Nightcrawler’s trivia section.
I am inclined to think he was Mystique and Azazel’s son.
I am not convinced that that Origins video games trivia counts as canon. But I do not think Mystique ever knew this was her son in either timeline. He must have been fostered by the circus family in the tie-in X2 prequel comic after the Hellfire club was massacred and he was separated from his mother. I have no idea though where the name Wagner comes into play in the films though. It could come from the circus, or Mystique could have used it as a fake birth record from an assignment, kind of like the comics.
Also, this would probably mean that Nightcrawler was not born blue in the film universe.
Assuming he was born in 1963, this would make him about 20 in Apocalypse, 29 in Dark Phoenix and 40 in X2, so the timeline works very well here!