The Home Office does not require employers to find out whether or not employees who began working for them by 30 June 2021 have applied under the EUSS.
So long as a compliant right to work check has been done, and provided there was no time limit on the right to work, you will be able to keep employing the EU national after 1 July 2021 without doing another check or seeing evidence of their status under the EUSS.
This is because you will receive the statutory excuse as a result of your compliant right to work check. As a result, if an employee hasn’t applied under the EUSS and has become an overstayer – such that they don’t have the right to work – you as the employer can rely on the statutory excuse as a defence to employing them illegally.
The exception to this is that if you know or have reasonable cause to believe that they don’t have the right to work after 30th June 2021 e.g. because they haven’t applied or their application was rejected – this will render your statutory excuse null and void and expose the business to the civil penalty of £20,000 per illegal worker as well as a criminal offence.