After updating my iPad to iOS 9, it crashed and I lost ALL apps that hadn't been backed up. So how do I access my Game Center if I've forgotten my Game Center nickname?
Your Game Center account is linked to your Apple ID. Your nickname is not important, as one can change it whenever he wants. Just launch Game Center and sign into it using your Apple ID and password. As far as Game Center data goes, it's probably lost unless you wish to restore one of your backups, if you have one.