However, May has stated her desire to also leave the European convention on human rights (and replace it with a UK owned Bill of Rights). This is not something that's happening as part of Brexit, and no bills have been presented before Parliament with this as a component or purpose.
I would assume that the government would take the pragmatic approach that it's better to focus on Brexit for now, and deal with the ECHR once Brexit is over with. However, I have no inside knowledge of this, and that's just my wild and unsubstantiated assumption.
So voting for Remain was also a vote to keep the UK locked in to the ECHR. Voting for Leave was also a vote to release that lock.
Also, the UK is signed up to the ECHR which theoretically guarantees a right to privacy. It's sort of like the US Bill of Rights except useless, because it was drafted by Europeans so every right has a giant get-out clause. In this case the so-called "right" to privacy exists only as long as it doesn't conflict with the "needs of a democratic society". That sort of thing crops up all the time in this document.
It's not sufficient to have a constitution. It must have teeth as well.