The National Theatre, London: https://www.nationaltheatre.org.uk/

Located on the south bank of the Thames, the Royal National Theatre features three performance spaces: The Olivier, The Lyttelton, and The Dorfman.
The Royal Shakespeare Company: https://www.rsc.org.uk/
2019 Summer Season: https://www.rsc.org.uk/news/summer-2019-season-announced

Located in Stratford-upon-Avon, birthplace of William Shakespeare, the RSC is a repertory company dedicated to producing and reinterpreting classical plays
The Royal Court Theatre: https://royalcourttheatre.com/
2019 Spring/Summer Season: https://royalcourttheatre.com/Season/spring-summer-2019/

Since the mid-20th century, the Royal Court has promoted the work of new playwrights.
Shakespeare’s Globe Theatre: http://www.shakespearesglobe.com/

An architectural recreation of the original Globe Theatre in Southwark, Shakespeare’s Globe offers a variety of performances staged on its traditional Elizabethan-style stage.
National Theatre of Scotland:

The National Theatre of Scotland prides itself as being a “theatre without walls,” meaning they stage productions in found spaces all over the country
The story of how they got started here: “Room to roam: how Scotland’s vagabond national theatre broke free” by Mark Fisher