the LAPL exists under EU law. For it to be valid next year it will need validating under the U.K. ANO and will still be a sub ICAO licence. It will need a mutual agreement for it to work outside the U.K.
European regulations [the correct term] is not law until they are adopted by each individual state. The UK parliament does this by statute which enables it to have effect without requiring a long protracted parliamentary procedure. The LAPL together with all other EU regulation and licences have become law in the UK. All this leaves the individual and an organisation to continuously refer to both the EU regulation and the ANO for detail.
Parliament recently passed legislation to revise all such statutes with the main purpose of removing all reverence to the EU (including by default EASA). This was made controversial for different reasons but was passed. Norway is a member of the EEA and contributes financially towards the EU and like Switzerland has adopted much of the EU legislation by various agreements.
you are also confusing EU membership and EASA membership.
EASA is an agency of the EU. All EASA regulation is first developed through the EU Commision before being passed or rejected by the EU Parliament.