"Right to study" checks for unsponsored EEA and Swiss nationals have been the subject of much recent discussion. Student sponsor guidance document 2 provides some information about what sponsors must do from 1 July 2021 (copied below). Note that Appendix D has not yet been updated.
If you are a student or a prospective student, your education institution is likely to ask you for evidence of your immigration status. This is because the Home Office requires them to check that you have permission to be in the UK that allows you to study. Some institutions will ask you about your status, even if you started your course before 1 July 2021. Although the Home Office does not require them to do this, it is to ensure that you are aware of the need to apply for permission so that you do not encounter difficulties in the future. The deadline for most EU settlement scheme applications is today, though late applications are possible - see gov.uk.
"Right to Study Checks for EEA nationals
2.9 The normal policy set out in Appendix D on checking documents applies to all sponsored students, including those from the EU and EEA. From 1 January 2021, EU and EEA nationals entering the UK to study, who are not eligible for the EU Settlement Scheme will need permission on the Student or Child Student routes.
2.10 We expect sponsors to start checking, from 1 July, that unsponsored EU and EEA nationals who enrol on a course of study on or after 1 July 2021 have an immigration status that allows them to study in the UK subject to Covid-19 concessions on record keeping
2.11 We do not require sponsors to retrospectively check whether unsuspended EU or EEA students who enrolled before 1 July 2021 have an immigration status that allows them to study. In these cases it is the responsibility of the individual concerned to ensure they have the correct status and we will not take compliance action against a sponsor who has enrolled such an EEA student before date 1 July 2021 where the student has failed to obtain the correct permission."