The result of the recent referendum in the UK will have no effect on the implementation of GDPR here. There is no fixed date for the UK to leave Europe and until Article 50 of the Lisbon Treaty is instigated to trigger the exit process, there is no knowing at what point the UK will leave Europe.
GDPR however has already been passed – and passed as a Regulation – which means that the UK has no power to stop it being fully implemented on May 25th 2018 along with the rest of Europe.
Clearly, the UK will already be bound by the terms of the GDPR by the time the UK exits the EU. This means that the ICO, UK Courts and all affected entities will have already have put in place the requisite infrastructures to comply with the GDPR.
Certainly the UK Government would be free to abandon the GDPR after exit but why would they? It would be more expense at a time when the country will be looking to save money and it would certainly not in the interests of the UK, the UK citizens or the interests of UK-based data controllers and data processors.