Game saves not syncing to iCloud

Hi. I've been playing Arcade games on iPhone, and wanted to keep playing on my other devices (iPad and Apple TV), but the save data has not come across. I'm signed in with the same Apple ID for iCloud and Game Center on all devices, and on my iPhone I've made sure the games listed in iCloud are all toggled on. But when I look in my iCloud storage, none of them are there. Am I missing a step?