Once you have two-factor authentication enabled, you'll be asked to provide your second factor during login. The two-factor authentication codes are securely stored in the Google Authenticator app on your phone. You don't need to wait for any texts or notifications (in fact there aren't any). Just open the Google Authenticator app and the code will be immediately displayed.
Simply type in the code (without spaces) while it's displayed in blue. If it turns red the code is just about to change -- the codes change every 30 seconds, so just give it a few seconds and you'll get a new code.
If you switched to a new phone and did not use the Export Accounts screen to transfer the codes to your new phone, or if you deleted the Google Authenticator app from your phone by mistake at some point in the past, then you will have lost the codes. In this situation use your backup code as your second factor, then immediately go to your security settings page to disable two-factor authentication and re-enable it for your new phone.
If you've lost your backup code as well as the code from your phone, we're on hand to help. Just call us on 01642 688 088 during office hours so that we can help you regain access to your account.