Changing the login email on a Railway account is something we're unable to do on our end. Railway authenticates through external providers, and modifying which identity an account uses to sign in would bypass the security boundary those providers enforce.

Since you're currently logged in, you can continue using your account normally through that session. For the underlying issue with Apple Private Relay not delivering codes, you would need to resolve that through Apple's side (e.g., adjusting your Private Relay or Hide My Email settings so codes are forwarded to a reachable inbox). Once the relay address is receiving mail again, login will work as before.