From what I seen, I saw that it wasn't because of a setting (at least it wasn't for me). It was because of on your registration page, your 'Create Account' button doesn't seem to work and won't redirect the page to the registration page (since you didn't explain hehe...). As anyone installed something or changed something on the registration template? I know you said it hasn't been touched but we need to make sure that nothing has been touched or messed with, even if it's a modification, template change etc., just want to cover the base here.
Edit: Those who switch to the mobile version will be able to register normally so my best guess would someone has messed with the registration template.