I rather expect that the result is going to be an under-the-rug reintroduction of the urinary leash. Females will simply not use those services, and will stay dirty/sweaty/covered in chlorine until they can get to the safety of their own home in order to bath and change. They'll hold their bladder until they get home, or until they are certain there's nobody else in the toilet. And they'll always go in groups, never being alone in those areas.
It will result in limitations on how much females can take part in society.
Fortunately, I don't think it will come to that.
The way I read the political landscape around this issue is that very few people are actually affected by this in any meaningful way, so what is happening is people are drawing up sides based on ideological alignment rather than practical concerns of day to day life. Some of them are traditionalists who see transgender concerns as a threat to traditional values. Others are people who are seeking a way to stand up for an oppressed minority. It's like a left/right proxy war. Some real people are actually being caught up in this, like the ones I always cite in Washington with Colleen Brenna, or the Palatine schoolgirls, but, honestly, there aren't very many people affected. As long as that number remains small, people won't pay much attention. The majority of people will almost entirely ignore the issue, and not understand what's actually happening. The rest will fight their fight on principle, without paying much attention to the reality where it exists.
In the scenario you described, it would have to mean that a significant portion of the populace was actually affected in a practical manner by this, and if that were to happen, I think the majority would stand up and say enough was enough.
I could be wrong about that. Maybe the majority will decide that it's ok to have the transgenders around and sharing the bathrooms and locker rooms. If that's the case, then your scenario also won't happen, because women will just give in and decide it's not worth the trouble.
I don't see that happening, either, because I think that a few extreme cases go a long way in influencing public opinion. I think the majority of transgenders are pretty harmless, and if you happen to belong to a gym with one of them, I think a lot of women would shrug their shoulders and deal with it, sort of going back to the "old days" when everyone looked the other way. However, one Jonathan Yaniv or Colleen Brenna goes a long way. I think that what will happen is that, if transgender acceptance becomes commonplace in changing rooms, there will be a few high profile problem cases, and that will be enough to turn public opinion in the opposite direction.
So, I don't see your scenario taking place, but mostly because people will see it go slightly in that direction, and will put a stop to it. i think there will be some more cases like the British Home Office where someone puts forward the new order, thinking that everything will be just fine, and will be surprised that everything will not be fine, and while the people who put those policies in place mutter about "antediluvial" attitudes, voters will put people into office who support segregation by sex, in the traditional manner, with additional privacy options, and separate changing rooms recommended for transgender people, where it isn't cost prohibitive to do so.
