Is it obligatory to repeat the ritual bath (ghusl) after doubting the utterance of 'Bismillah' (in the name of God) and the washing of the hands three times before washing the private parts and the area of impurity?

Your ghusl (ritual bath) is valid, and you are not required to repeat it. Doubt after the completion of an act of worship is not taken into consideration. Saying "Bismillah" (in the name of Allah) and washing the hands at the beginning of ghusl are sunnah (recommended) acts, not obligatory ones. The obligatory components of ghusl are the intention (niyyah) and washing the entire body.