I am very modest also which is a major part of why I like water birth. The water really distorts sights, sounds, and emissions from down there. I wear a black (def not white!) tank for the water. When I get out I'm immediately stripped to get cold, wet clothes off. There's really no getting around this in my mind.
My (then) 12yo son was at my #5 birth and I felt very comfortable with him there and his father guiding him to hold the baby after the birth so I could get out, get checked, stripped, and redressed. To put this in perspective, my son has never seen me in less than shorts and a sleep tank top ( def not bra and panties) and I swim in a rash guard and capris swim pants.