The UK will have to maintain an adequate level of protection for the rights and freedoms of data subjects in relation to the processing of personal data if it wants to continue trading with EU businesses. What is or is not adequate will fall to be determined by the EU. So while the EU's new General Data Protection Regulation will cease to have direct effect in the UK after Brexit (it is due to come into force in May 2018), the UK is probably going to have to enact something very similar if it wants to continue to trade with the EU. (And even if it doesn't, UK businesses could still fall foul of the extraterritorial provisions of the GDPR which apply to any business trading within the EU, wherever in the world they may happen to be located.)
This is reinforced by the Information Commissioner's Office statement on the matter:
"The UK will continue to need clear and effective data protection laws, whether or not the country remains part of the EU.
The UK has a history of providing legal protection to consumers around their personal data. Our data protection laws precede EU legislation by more than a decade, and go beyond the current requirements set out by the EU, for instance with the power given to the ICO to issue fines. Having clear laws with safeguards in place is more important than ever given the growing digital economy, and is also central to the sharing of data that international trade relies on."Intellectual Property
UK trade marks, registered and unregistered designs and copyright will continue to be protected under existing UK legislation (much of it harmonised with EU law). There may, however, be some uncertainty as to the exact scope of protection, given that considerable case-law has developed from answers to questions referred by UK Courts to the Court of Justice for the European Union. Precedents on the test for originality in copyright law, and unfair advantage in trade mark law, for example, may all be open to new interpretation.
UK businesses will still be able to take advantage of the European Patent Convention (which is nothing to do with the EU) to simplify patent applications in individual European states. They will also still have access to the new pan-European Unitary Patent (if it comes into effect at all, assuming the technicality of the need for ratification by the UK as an EU Member State can be overcome). However, the protection of the Unitary Patent itself will no longer extend to the UK, which may damage its attractiveness in the eyes of many rights holders.
Perhaps the biggest immediate concern for businesses is the status of the Registered Community Design and EU Trade Mark (previously known as the Community Trade Mark). At the moment holders of these EU-wide registrations have protection for designs and trade marks respectively in the UK as well as all other current EU Member States. Once the UK brexits, however, then absent any other arrangements, protection in the UK from these registrations will lapse.
The consensus seems to be that transitional arrangements will be put in place, allowing businesses to convert their existing EU-wide design and trade mark registrations to national registrations covering the UK, with the existing registrations continuing to cover the remainder of the EU. While this may indeed be the most likely scenario, it is far from a guaranteed outcome. If you don't have existing national registrations, and want to "take control" like a Leaver, you may want to consider making new applications for UK registrations now if national protection is absent from current portfolios.
On the other hand, just as with the wider question of future EU relations, you may just want to hold your breath, hope for the best and see what happens.