I can actually tell you exactly how much time it used to take.
Pre-mega costuming warehouse
We had costuming more or less behind Mr. Lincoln. It had 20 windows -- 10 for guys and 10 for girls. Each one usually had a line of 2 or 3 people, and it usually took 10 minutes or less from when you got in line to when you walk away with your costume for the day. Then I had to walk to your lockers, which were mostly behind Space Mountain. I had a permanant locker. Then, after changing, I would have to walk from my locker to my office, which was behind Splash Mountain. So I always budgeted 30 minutes to make sure I had time to get my costume and change. Sometimes I'd get to the office 10 min early and sometimes I'd be running to make it on time.
They built a new costuming warehouse to handle all of the extra costumes for DCA in addition to Disneyland in 1999. It is the big brown building you see to the east of the Dog Kennels. This building also has lockers, but now they are day lockers and so no one is assigned a permanant locker any more. They are a little further away, and even though there are now all of the DCA cast members that need to get costumes as well, they still have the same number of windows as the old building (although giant counters, as if they expect there to maybe be more than one CM per counter helping people, but I never encountered that). It's quite a bit further away then the old lockers, so between longer lines at costuming and having to walk a little further, I added an extra 10 minutes and would try to budget 40 minutes to get my costume and have time to change and make it to my office.
So pre-DCA, I would have to add half an hour before and half an hour after my shift, and post-DCA, I would have to add 40 minutes before and 40 minutes after my shift. When we would turn in the costumes at the end of the night, we couldn't just drop them off -- we had to wait in line to have a CM check them in so that we'd be allowed to get a new costume the next day.
So today, a CM saves 80 minutes a day, minus the time they have to spend the one or two times a week they do go to costuming in order to turn in all of their old costumes and receive all of their new clean costumes.