When you are obsessed with another culture, you tend to ponder its nature in every detail: what did people wear, how did they cook, where did they go for recreation and how long did they live. Everything is of interest including bathing rituals. I think about modern man’s focus on cleanliness (witness the plethora of products on market shelves) and wonder how people lived with dirt and grime—and no washing machine! Wow. They had to rub it out at the river’s edge on themselves or their clothing.
I know they had amazing baths in ancient Rome and no doubt some type existed in Egypt. After all, they were an advanced society with ample access to water. I am sure that it was environmentally friendly according to today’s standards given the natural resources used. Now our water heaters eat up energy and the world is evolving toward tankless models to stem the tide of waste. If people are going on paleo diets, why not adapt old bathing procedures and be mindful of the need to go green.
Heating water was no doubt an issue for the Egyptians who had to use fire instead of an electric or gas appliance. We can’t do that safely today although in the late 19th century, this was the mode of operation, especially in the Old West. It was a long process so family members had to use the same water until it turned cold. I imagine this is true for the Egyptians, except maybe Cleopatra, who got her own bath all to herself, complete with essential perfumes and oils.
Since you can’t replicate the old ways of the ancient world, you can only do what you can these days to be judicious with energy usage and feel more responsible. Taking shorter showers and smaller baths is certainly useful in this regard. Tankless water heaters are the major methods of choice to save money on utilities (source: https://tanklesscenter.net/environmentally-friendly/). If we all got a new unit, imagine the savings worldwide. You are right in line with the Egyptians who certainly went tankless by necessity! Ha! So we can emulate them in spirit by not storing water at a high temperature, even when not in use. Like them, we use water when needed in the moment. In fact, there is no reason to do otherwise.