The Embassy of Hungary in London held a discussion evening with British officials where Tamas Stenczel and Sara Mandoki participated to get information regarding the implications of Brexit from first hand. Please find attached the summary booklet with all the information currently accessible.
The UK will leave the EU on 29 March 2019 and will move to a transition period until 31 December 2020. If you arrived to the UK before the end of this period, you will be able to apply for staying here and can finish your studies in peace with fees and student loan as now. The application process will be simple and cheap (<£100). There is no need for you to do anything now, but it is advised to do the application as soon as possible after it will be open.