I am researching this for a fantasy story so that is why I am not particular on the exact time period or place. I am specifically interested in bathhouses with water sourced from hot springs, whether in ancient times or prior to the 18th century, but again if there is no data for it then I'll take anything!
How could you drain and fill baths without electricity or some type of engine? Was it possible to pump water by hand or possibly by steam power to raise it up so that gravity could then be used to move the water to a bath/etc? What was the process of tapping into and diverting a water source like prior to the 18th century?
I have found information on a variety of pumps and methods used to transport water prior to widespread mechanization and electricity. One beginner resource I stumbled upon was Evolution of Water Lifting Devices (Pumps) over the Centuries Worldwide by Yannopoulos et AL (https://www.mdpi.com/2073-4441/7/9/5031). (on mobile, pardon my formatting). Online resources for Roman bathhouses seem scant as well when it comes to the nitty-gritty of how they moved the water from place to place (beyond the aqueducts providing gravity fed water).
I hope this is appropriate for this forum. I have learned a lot from lurking and thought I would ask about this issue. If this question is too vague, I can remove it and think of a better way to poise my question. Thank you all!
Not so directly relevant to your question: I wrote a brief notice a couple of weeks ago on how the medieval Icelanders supplied the hot water from springs to the bathhouse in this question thread, though I had not found any literature on such plumping mechanics in pre-modern times as you mentioned in OP.
I also hope that someone knowledgeable with Islamic, especially Persian and Turkic bathhouse may post and tell us how they filled bathhouses in pre-modern times since especially the Persians were very famous for their technology of irrigation as well as the bathhouse culture.