We are still not clear what the impact will be for EU citizens coming into the UK after that time for travel, study or work. English UK, the trade body for English language schools in the UK, has an updated page on Brexit and how it is likely to affect students. You can find more updated information here
Some things are known:
The UK will remain part of the Erasmus+ programme until it finishes in 2020, Brexit will not impact on the UK’s position as a programme member and funded projects with UK partners will be unaffected. More info is available from the UK national Agency.
Entry to the UK
This will remain unchanged until 31st December 2020. EU/EEA citizens will still be able to enter the UK without a visa and can enter with their ID cards until that time. Students planning to stay for more than 3 months will need apply for Permission to Remain from the UK Home Office.
Flights into the UK should remain unaffected (i.e. air travel to/from the UK should not stop) even in the case of a No Deal Brexit but there is likely to be some disruption. Please check with your travel agent or airline if you are concerned about this.