The singer had been in rehearsals for a musical version of Dolly Parton’s most famous film, in which she was set to play Violet Newstead.
“It breaks by heart to say I am having to take some time out and won’t be able to continue with my role in 9-5,” she wrote on Instagram.
“Due to a nasty fall I have only gone and fractured my wrist and got 10 (yes 10) stitches in my chin.
“Following my doctors orders – they have said I will be ok but as they say in the theatre ‘the show much go on’ and I wish the rest of the fabulous cast an amazing opening night and I will be back on that stage for 9-5 before you know it.”
Producers of the show said Redknapp’s replacement would be announced shortly, and that anyone wishing to exchange their tickets to when she would appear in the show should contact their point of sale.
They said she would likely be returning to the show in March or April.
“The recovery time recommended means Louise is no longer able to properly rehearse the show in the remaining time left before opening,” they said in a statement.
“Together we have made the difficult decision to delay Louise’s start date in the show.
“Everyone at 9 to 5 wishes Louise a speedy recovery and we look forward to welcoming her back.”
Redknapp was due to appear on stage alongside Amber Davies, Love Island’s 2017 winner, who will play Judy Bernly in the show.
The performer wished Redknapp a speedy recovery on Instagram.
“One of my favourite humans,” she wrote. “I wish you a speedy recover (sic) and can’t wait to watch you be a leading lady. See you Tuesday.”
The production will run at London’s Savoy Theatre from 28 January.