An application to register must contain any address where the applicant has ceased to reside in the 12 months prior to the date of the application and, where that address is not in the UK, an indication of whether that person was registered as an overseas elector during this period.1
You have a duty to notify the relevant ERO where you have been given a previous address in the UK.2
This process is automated through a link between your EMS system and the IER Digital Service and the notification triggered once you have allowed the application.
However, a person may be entitled to be registered at more than one address (e.g. students). If on their application they claim that they are entitled to be registered at more than one address, you should contact the elector and the other ERO to establish the details of their residence.