Aimee, 21, had been set to begin her run with her mother in the West End production of the show on Tuesday.
The spokeswoman, in a statement, said the family was flying to Los Angeles over the weekend so that Aimee could have "further treatment and rest." There were no details about her illness.
"They were both very much looking forward to appearing in the show, but Sharon's focus now has to be on the family and, more importantly, Aimee's health," the spokeswoman said.
Aimee was the only family member not to appear in the MTV reality show, "The Osbournes," that catapulted the family to fame.
Last week, Ozzy and Sharon Osbourne had to flee their English country mansion after a fire broke out in the living room. The blaze was the latest in a series of domestic mishaps to befall the former Black Sabbath frontman and his wife.
Ozzy unsuccessfully tackled a thief as he raided the house and stole a large amount of jewelry last November. And in December 2003, Osbourne was in a coma for eight days after breaking his collarbone, several ribs and a neck vertebrae in a quadbike accident on the grounds of the home.