In a situation where a mikva is used at different times by both men and women, is there a need to change the mikva water between uses?
In order to keep Mikvaos functioning the whole year round, the Mikveh water that is used for dunking is not the same water that is used for the Mikva itself. What is done is two processes called hoshoka (Kissing). The mayim shoevim (non mikva water) is made to connect to the Mikvah water. Once that happens, then all the waters become kosher for immersion.
In general, a Men's mikva, they often just do one hoshoka. In a women's mikvah they do things l'chumra (stringently) and do hoshoka in both ways. A men's mikvah is only because of takanas Ezra (the edict of Ezra) and so is Middos Chassidus (a better thing to do if you can). Women's mikvah is for Tahora for Tahras Mishpacha - an issue of D'oraissa (Torah prohibition), and so we're more machmir (stringent). It would seem therefore that a women's mikvah can also be used for a men's mikvah but not vice-versa.
As regards the actual change of water of the place of dunking. It would probably be a matter of practicality and cleanliness. Mikvah water can get dirty and murky easily, unless of course, they didn't do a proper neshika. (see here). So the water would have to be changed for women. Since we want women to do the mitzvah, we make sure that everything is setup accordingly for her benefit.
The Iggros Moshe writes that given that it would be uncomfortable for women to use the same mikvah as men, it would be better to move the men's mikvah away (See Iggros Moshe, Yoreh Deah 2, 90). Thus, we see that any discomfort on the part of a woman is enough to stop the men's mikvah entirely. There would be an implied necessity to change the water after the men's mikvah.