Oh I bathed my little one as per normal even with the umbilical cord. It did touch water a little. I avoided the area while soaping, and dabbed it dry after the bath. I cleaned it with a moist cotton ball (just warm water) and let it dry. Even after the cord fell off, while there were still some scabs, I continued to clean it with the damp cotton ball daily, till everything had fallen off. Everyone has their own style I guess.
We towel clean our girl (no bath) before her umbilical cord drop. Used room temp (boiled water) to clean her cord area with cotton wool. After the umbilical cord dropped, we put her in a bath tub & bath her like normal. Just make sure we dry the cord area each time after bath.