There are essentially two ways to leave the EU. 

The UK could agree a Withdrawal Treaty with the EU, meaning nothing would change during an extended transition period ( to December 2020) after Brexit day. 

Nothing much will change for UK Expats as EU citizens they will continue to enjoy freedom of movement. 

With no deal, after Brexit date UK Expats would be subject to the current rules when relocating to any country. 

Generally this means up to a 5 years temporary residency visa. 

Often these can be renewed. 

After 5-7 years typically permanent residence or citizenship can be applied for based upon the rules of the country. 

Different countries have different barriers to entry. 

Contact us for more guidance in your instance. 

So yes UK expats will revert to the the Cyprus rules pre Cyprus joining the EU in 2004. To apply for up to a 5 year temporary residency visa.

Contact us, ProACT EU Brexit for Expats Experts, for a free review and guidance.