I guess it depends on the country. In Poland my experience is that every time EU passes some regulation Polish parliament passes the corresponding bill implementing it. So even if we left EU tomorrow those bills will still be in effect.
Also, I'm not sure about this "immediately enforceable" part - I recall some cases where member states delayed implementing EU laws for years, sometimes ending up being sued to European Court of Justice.
Yes, the UK has the DPA-2018 legislation which basically implements GDPR in the UK. [1] Its application is supervised by the ICO. Subject Access Request can be issued to pretty much any public institution / company [2]
"You have the right to find out if an organisation is using or storing your personal data. This is called the right of access. You exercise this right by asking for a copy of the data, which is commonly known as making a 'subject access request"
GDPR directly doesn't apply to the UK anymore (except for organisations that handle data of EU citizen, of course) but the UK chose to enact the GDPR into UK law via the Data Protection Law of 2018, which is aptly dubbed "UK GDPR".