Haque, who also appeared in Coronation Street, was diagnosed with cancer before Christmas, the BBC reported.
She was suffering from pneumonia and was understood to have developed a clot on her lungs.
Michael Grandage, who directed Privates on Parade, said he and her fellow cast members were “devastated”.
Haque dropped out of the production about two weeks ago.
Born in Portsmouth, she became a familiar face on television during a seven-year stint as a presenter of MTV Asia.
A talented singer and dancer, she moved to Mumbai in 1997 and starred in a number of films.
On her return in 2002, she starred in Andrew Lloyd Webber’s Bollywood-inspired musical Bombay Dreams and took a leading role in the musical adaptation of The Far Pavilions.
More recently, she starred in Wah! Wah! Girls at London’s Theatre Royal Stratford East and Peacock theatre.