If the person who posted the question is becoming a Baal Tsuvah – there is no issue. No one will challenge that he is Jewish, and even if the did it would be very easy to have a conversion (since conversion of someone who is already frum and discovers they might not be Jewish is extremely uncomplicated).
If the case came before Beis Din in a situation where the woman has a child from the non-Jewish second husband, and that child wanted to become a Baal Tsuvah – most Betei Din (in practice) would be super-critical in looking at the validity of the first marriage and the validity of the conversion in order to avoid ruling that the hypothetical child of the second marriage is a mamzer. Excluding politicized Betei Din in Eretz Yisrael, our courts and rabbanim have always displayed great intelligence and “thinking like a lawyer” to resolve such problems (based on actual cases in printed shailohs and tsuvahs, not based on theoretical treatises).