Britain will cease to be a member of the EU by March 29, 2019, according to the Lisbon Treaty, which counts on the entire exit process being completed within 24 months. At that point, UK citizens will no longer be EU citizens, as they are now. Also, EU citizens living in the UK will most likely not continue to enjoy the same rights that they currently do.
British officials are now saying that there will be a different approach to those Slovaks who were already living in the UK “before Brexit” – assumed to mean the date on which the UK formally leaves the EU – compared to those who come afterwards, but it is not yet clear what exactly that will amount to.

Brits that have lived in Slovakia and Slovaks that have lived in the UK long enough to be eligible for citizenship might therefore be considering obtaining a passport from their country of residence.
The Slovak citizenship procedure can take up to two years. People who have permanently resided in Slovakia for eight years may apply for citizenship (or five years if married to Slovaks and living in Slovakia).

Both Slovakia and Britain recognise dual citizenship.
“The law does not require the applicant to give up the state citizenship of his or her home country before he or she is granted the state citizenship of the Slovak Republic,” Lucia Kirinovičová from the Interior Ministry’s press department told The Slovak Spectator.
The matter is more complicated forSlovaks living in the UK who might be thinking about becoming British citizens. After the citizenship law was amended in 2010, Slovak citizens who request and are granted the citizenship of another country are required to report this fact to the Slovak authorities. In most cases they will then lose their Slovak citizenship.

“The law is effective also for our state citizens living in Britain, regardless of the length of their stay there,” Kirinovičová noted.
Exceptions apply to those who are married to British citizens or whose parents are British citizens. People in these categories can normally retain their Slovak citizenship even if they become British citizens.