The response time will vary. If you get a CM who feels your situation is valid, he/she will pass it to a Guest Relations supervisor who usually calls by phone in order to further discuss issue. If not by phone, he/she will email to tell you that they have passed your email on to the appropriate department, so you'll wait for further instructions.
I would give it at least 7 days, and if you don't hear anything, try again.
I am known as an amateur consumer advocate around here, and I've contacted thousands of companies in my day. Disney is pretty good about responding and doing something about the situation. Only once in awhile, did an email get ignored (had a merchandise credit issue), and I sent email again, and it was rectified right away.
It helps to have a short and simple email that gets right to the point - I'm an English teacher so I can attest to the fact that most people only quickly read over items, and customer service reps are the worst at that, since they do it all day long. So, if you have some key points in there about the fact that an error happened beyond your control, and due to that error on Disney's part, it cost you, the guest, $20.00 in related exchange fees, then I'm pretty sure you'll get a response with a gift card offer or something similar to make up for the loss of money.
Let me know how it goes, and I can write an email for you if you like!
Good luck, Tiger