The family musical, which reimagines the classic children's tale, will play mornings and afternoons from 5 August to 6 September.
Alison Jiear was Olivier-nominated for her performance in Jerry Springer: the Opera, and her other stage credits include On the Town and Shoes. She impressed this year's BGT judges with her rendition of "You'll Never Walk Alone" from Carousel, but was knocked out in this week's semi-finals.
Simon Webbe is best known as a quarter of the boy band Blue. He has appeared in Sister Act at the London Palladium, and has also been a contestant on Strictly Come Dancing. Gareth Gates was the runner up in the first series of Pop Idol, and has gone on to have a successful music career. His theatre credits include Joseph and the Amazing Technicolour Dreamcoat, Les Miserables and Legally Blonde.
The rest of the cast includes Leanne Jones (Hairspray, Molly Wobbly's Tit Factory) and Taofique Folarin (The Lion King, Avenue Q).
Stiles and Drewe said of their latest musical: "We love surprising and bewitching our younger audiences, some of whom will be visiting the theatre for the first time. We take it as a great responsibility to encourage and nurture audiences of the future, so we try to ensure their first experience of a musical is a memorable one."