IP rights in the UK are all influenced and moulded to a greater or lesser degree by European law.
The different rights exist on a spectrum, from those that are unitary pan-European rights (such as Community Trade Marks and Registered Designs), to those that are heavily harmonised (for example national trade mark rights under the Trade Marks Directive) to those that have common features, but also national differences (such as copyright and patents).
The task of unpicking the implications for those different rights in the event of a Brexit would not be simple, but presumably the current legislation would remain exactly as it is for a considerable period. One immediate problem would be to assess whether pan European rights (Community Trade Marks and Registered Designs) would continue to offer protection in the UK post-Brexit. Presumably they would not, but to avoid huge disruptions to business it can only be assumed the UK's Intellectual Property Office would have to offer a system of "grandfathering" over the "UK element" of those European rights into separate analogous UK rights. Brexit would also almost certainly mean the end of the UK's eligibility to take part in the proposed new unitary EU patent and its planned role as a host of one of the central divisions of the proposed Unified Patent Court.
The limits of IP rights in the UK have also for many years been defined by the interplay with EU competition law. So, for example, trade mark licences to distributors could not prohibit those distributors from engaging in passive sales outside of their territory but within the EU and the act of placing IP-protected products on the market in one member state is deemed to "exhaust" the proprietor's right to object the subsequent free circulation of those goods within the EU. If UK IP law and EU competition law is to converge, then UK right holders may be able to control more tightly the circulation of their products on the wider market.