Google tells me the average bath uses 35-50 gallons of water, while the average shower uses 2 gallons per minute. Which puts the breakeven point at say 15-25 minutes.
Although if you live in the midwest like I do, we have infinite water. Use more water? it goes back to lake Michigan and recycled. The gas to heat the water is the only "waste", but you could offset with a solar panel.
Creating potable drinking water from natural sources requires a little energy. You will probably notice it in your water bill if you start showering twice as long.
While it's twice the amount of water, the water is not as dirty (when you shower twice as long you may not be twice as dirty and may not be using twice the amount of shampoo).
Is that reasoning unrealistic? I don't know a first thing about water treatment.
For drinking water, we typically process relatively clean sources of water to begin with, not wastewater. It will vary by region. (I don't think anywhere actually starts with wastewater rather than some more convenient and clean local source, outside of fiction like "Dune.")
To take a local example I am familiar with — Seattle gets most of its drinking water from the Cedar River watershed (rainwater). It is a relatively pristine source. Wastewater, including shower output, is treated and released into Puget Sound[0] (the ocean). The only way that ocean water makes it back into the drinking supply is via ordinary evaporation and rain.
So the treatment required on the second quantity of water is exactly the same as on the first, and you get linearly increasing cost.
or you do it like in japan : they have a bathtube filled with water that you reheat when you want to have a bath, but you actually clean yourself before going inside.
