Currently the UK remains a member of the EU and, therefore, there is no immediate change to the position of EEA nationals in the UK. Therefore, students and their family members can continue to exercise their residency rights in the UK.
The Home Office has published a statement providing information on the status of EU nationals currently in the UK. This includes some information about the right of permanent residence and UK citizenship. The Home Office has confirmed that from 1 August 2016 applications for naturalisation as a British citizen which do not include a document certifying the right of permanent residence will be refused.
We do not yet know exactly what the future implications for residency will be for EEA nationals and their family members who wish to come to, or remain in, the UK to study. This will depend on how the UK and EU might agree to deal with transitional arrangements about the position of current and new students. Any negotiations are likely to take some time, as there are many aspects of the UK’s relationship to the European Union which will need to be considered. Migrants' Rights Network hosted a Q&A session to help provide some information and advice to EU nationals in the UK. In addition, the Free Movement blog has posted a Brexit Q&A podcast on it's forum to help answer questions which have arisen since the UK's decision to leave the EU.
We will continue to publish relevant information here as soon as it becomes available.