The prizegiving, which will honour the best performances and plays in U.K. theatres over the past year, will take place at London's Old Vic theatre on 13 November (16).
Close's performance as fading film star Norma Desmond in Sunset Boulevard has earned her a Best Musical Performance nod, while MCKellen and Branagh will battle it out for the Best Actor trophy for their performances in Harold Pinter's No Man's Land and The Entertainer. They will be up against Ralph Fiennes (The Master Builder) O-T Fagbenle (Ma Rainey’s Black Bottom) and James MCArdle (Platonov).
Noma Dumezweni has also landed a Best Actress nomination for her role as a beauty brand executive in Linda - a part she took on after replacing Kim Cattrall, who was forced to pull out of the play. Noma has since gone on to star in Harry Potter and the Cursed Child, the stage show continuing the story of J.K. Rowling's boy wizard into adulthood.
She will fight it out with Helen MCCrory (The Deep Blue Sea), Sophie Melville (Iphigenia In Splott), and Billie Piper (Yerma), for the top award, which is named in honour of the late British actress Natasha Richardson.
Harry Potter and the Cursed Child is up for Best Play, alongside Suzan-Lori Parks' U.S. Civil War drama Father Comes Home From The Wars and Annie Baker's Pulitzer Prize-winning play The Flick.
Funny Girl, Groundhog Day, Guys And Dolls, Jesus Christ Superstar, Our Ladies Of Perpetual Succour, and Sunset Boulevard are all contenders for Best Musical.
One other notable nominee is John Malkovich, who is up for the Best Director prize after staging the play Good Canary at the Rose Theatre in Kingston upon Thames, England.
The 13 November (16) awards ceremony will be hosted by rock icon Sir Elton John and Evening Standard owner Evgeny Lebedev.