At age 82, Broadway director and producer Harold Prince has 21 Tony Awards—more than anyone in the history of the business. Over the past six decades, he's produced hits from "West Side Story" to "Fiddler on the Roof," and directed shows like "Cabaret," "Sweeney Todd: The Demon Barber of Fleet Street" and "The Phantom of the Opera."

His next project: a new musical comedy based on an obscure Austrian novel, set to 19th-century music and opening next month at a tiny theater in southeast London. Mr. Prince is betting that he...