Question: Somehow, I expect I know the answer to this already. Sigh. In a scene I am writing set in the Pump Room, I have described a marble fountain dispensing the waters, only to go, “Wait a minute, is that more Victorian than Regency in its description?” So . . . in the Regency, would the waters have been dispensed by servers at some kind of bar, instead of from a fountain with spigots?
Answer: I found this quote from Google Books. Hope it helps a little.
The Early Days of the Nineteenth Century in England, 1800-1820, Volume 1 – By William Connor Sydney
An interesting glance at the Bath of 1811 is afforded by a letter of a Canadian traveller: “On one side is the pump, where a woman stands and distributes old King Bladud’s waters to old and young, sick and ill. An old duchess of eighty and a child of four were both drinking the waters while I was there. I had ‘a glass; it is very hot and tastes very mineral. At one end of the room is an orchestra, where bands of music are continually playing. The company at the same time walking up and down in crowds, minding the music, but buzzing like merchants on-‘change.”
Also, I have a “volume” of notes on a variety of subjects (some 800+ pages), but I do not know from whom or which source this quotes comes: “In the centre of the south side is a marble vase from which issue the water with a fire-place on each side.”
In addition, I have read multiple novels where the author describes that the water was distributed from a fountain. Now, that does not necessary mean they were correct. It could be a Georgette Heyer kind of thing where some of what the author made up is now considered part of the “legend of the Regency.”
For a novel with detailed descriptions of bathing in Bath (which I have not seen elsewhere), try Carolyn Korsmeyer’s Charlotte’s Story, originally published TouchPoint Press in 2021. It is an Austenesque story about Charlotte Lucas, I have not tracked down the author’s sources, but she is a meticulous researcher and an editor, so I am confident she has completed her search for facts. There are quite a few descriptions of Charlotte Lucas involved with the water in Bath, not just the Pump Room.
To ease the strain of their relationship, Charlotte leaves her husband to visit the fashionable city of Bath with several women companions. The weeks there prove to be a time for self-discovery and freedom, and the marital frost begins to thaw. However, events in Bath result in an unfortunate, even calamitous, consequence.
Charlotte devises an audacious solution that combines bold connivance and compassionate duplicity, pursuing her hope of happiness with the wit and courage to seek it.
The ever fabulous Shannon Donnelly once shared this link: There are a couple of images that show the lady distributing the waters from behind a counter, some sort of actual pump behind her:
1798 –
1825 – https://bathartandarchitecture.blogspot.com/2018/08/beau-nash-statue-in-pump-rooms-bath-by.html
Jane Austen’s World has a wonderful piece on The Pump Room’s Little Known Facts
Every Woman Dreams... 
Wonderful post and very informative. Thanks!