US actress Sadie, 23, rose to fame after joining the cast of Stranger Things in series two as as Max Mayfield.
She grew up up as a "Broadway kid", making her stage debut, at age 10, in New York's famous theatre district as the lead in Annie.
But she'd "always dreamed" of doing a West End show, she said.
"To get to do that in one of Shakespeare's most famous plays under Rob's direction with Noah will be such an exciting challenge," she added.
"London theatre has this incredible energy, and I can't wait to be a part of it."
The actress appeared to be getting settled into London life when she joined New Zealand singer Lorde on stage at her concert at The O2 over the weekend.
Despite Romeo & Juliet being the first time Sadie will do theatre in London, she has revisited her roots in the US.
Last year, she was nominated for a prestigious Tony Award for her starring role in Kimberly Belflower's John Proctor Is the Villain.
She also appeared in 2022 Oscar-winning film The Whale, and this year's post-apocalyptic drama O'Dessa.