In terms of data protection, it is "business as usual".
The GDPR will continue to apply to the UK until the end of December 2020, which means that businesses in the UK that process personal data must continue to ensure that they are compliant with the Regulations. During this transition period, businesses which provide goods and services to those in the EU will not need to appoint a European representative.
Post-December 2020, it is likely that the GDPR will implemented into UK law as "the UK GDPR".
As for the Privacy and Electronic Communications Regulations ("PECR"), these will continue to apply irrespective of Brexit.
The UK’s post-Brexit transition period will pick up where the country’s EU membership left off, offering a temporary safe harbour while Brussels and London haggle over their long-term relationship.