It will be the national authorities who will have the responsibility. The U.K. government policy is to have equivalence in data regulation with the rest of the EU, so even after Brexit we'll be operating on the same rules, but it will be the national authority of the Information Commissioner to enforce those rules.
I think it's also worth mentioning that even though we have European standards for data management through GDPR, there are different requirements in different countries as well, and for different reasons. So Facebook in Germany runs to different rules than Facebook does in the U.K. because it has to comply with German legislation on hate speech, and therefore employs vast numbers of people to take down posts that would be in breach of Germany's data protection laws, and Facebook itself would be penalized if it failed to do that. So even within common European standards, you could still have quite different jurisdictional requirements in different member states.