I would go for "changing room".
"Fitting room" is definitely wrong. This is a rather old-fashioned term for the room at a tailor's, dressmaker's, or shop where you try on clothes to ensure that they fit.
"Locker room" is an AmE term. Internationally, it might confuse people, especially if there are no lockers in it. It might suggest that this is a place where things are stored. For example, a sports team might have a locker room where they change at a stadium, but also where they store their kit: sports clothes and equipment .
I've checked on some American sites, and they seem to use both 'changing room' and 'locker room' for public leisure facilities. However, the term 'locker room' may not be familiar to non AmE speakers, so I think it would be clearer to use 'changing room' in an international context.
"Dressing room" - your alternative which you suggested in a comment - is also wrong. "Dressing room" is used in a theatrical context - it's where performers put on their costumes and make-up. You also find dressing rooms in grand houses, where gentlemen and ladies dress ( or are dressed by their servants!). Again, not the right word for a public swimming pool.