If a Jewish man and non-Jewish wife have a child, and the mother completes her Orthodox conversion to Judaism after the child is born, must the child also convert to be considered Jewish, if they were raised Jewish from their time of birth?
A child born to a Jewish mother is Jewish,1 and one born to a non-Jewish mother is not. This is determined at the time of birth, which is why infant conversions are sometimes done when a woman is in the process of conversion. Changes in the mother's status after the child is born are not relevant. One source for this is Kiddushin 66b (in the mishna at the bottom of the page).
(I'm sure there are more sources; this information is widely taught and, so far as I know, not disputed. Here's one unsourced opinion from an Orthodox rabbi.)
1 Shulchan Aruch EH 4:19 discusses some special cases: the child of a female mamzer is a mamzer and the child of a Jewish woman and a non-Jewish man is kosher but defective for kohanim. (h/t DoubleAA)