Bonham Carter replaces Vanessa Kirby in the Left Bank series, while Daniels replaces Matthew Goode.
It comes after the SVOD service released the first image of season three of the royal drama, depicting Olivia Colman as Queen Elizabeth II.
Season three will launch in 2019. It begins in 1963 and will cover events such as the rise of the Beatles and England winning the soccer World Cup in 1966. Characters such as Camilla Parker Bowles will also start to emerge in the next season.
This comes after season two of The Crown secured 13 Emmy nominations last week including nods for Claire Foy and Matt Smith as well as the lead drama category.
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