The venue, which has been described as "Britain's seediest VIP club", was chosen by the composer for its similarity to the nearby Murray's Cabaret Club, where Stephen Ward first met Christine Keeler.
"I first got interested in this idea when I started reading about the trial of Stephen Ward," Lloyd Webber told the gathered media. "To this day you still can't get a transcript of the trial, which is completely extraordinary... I began to wonder who was this character Stephen Ward?"
He added that, although he remembered the Profumo Affair from his time at school, he didn't recall much more than the basic details.
"I then mentioned on [radio show] Front Row that I was intrigued by doing a musical about Stephen Ward, and the next day I got a call from Mandy Rice-Davies [who knew Ward and Keeler]. She met me and talked to me about Stephen, and how he became the scapegoat for a number of things.
"The police were instructed to get something on Ward, to turn him into an apology for what went on."
Speaking to WhatsOnStage.com, Hampton said the story "opened the door" on the swinging Sixties.
"Ward was so much at the centre of London society at the time, everybody knew him," he said. "His list of clients is barely believable - Gandhi, Churchill, Ava Gardner - these people were all friends of his... He's the impresario of the whole affair."
"We just followed the story where it took us - I've speculated, but I haven't invented anything," Hampton added.
The launch featured performances of four songs from the show, "Human Sacrifice", "This Side of the Sky", "Never Had it so Good" and "I'm Hopeless When It Comes To You". The latter was performed by Joanna Riding, who plays Profumo's wife, accompanied by Lloyd Webber on piano.
The principal cast comprises: Alexander Hanson (Stephen Ward), Charlotte Spencer (Christine Keeler), Charlotte Blackledge (Mandy Rice Davies), Anthony Calf (Lord Astor), Daniel Flynn (John Profumo), Joanna Riding (Valerie Hobson), Ian Conningham (Ivanov), Chris Howell (Murray), Ricardo Coke Thomas (Lucky Gordon) and Wayne Robinson (Johnny Edgecombe).
Other cast members are: Martin Callaghan, Kate Coyston, Jason Denton, Julian Forsyth, Amy Griffiths, Paul Kemble, Emma Kate Nelson, Carl Sanderson, Emily Squibb, John Stacey, Helen Ternent and Tim Walton.
See also: Our photos from today's launch event. And keep posted for video coverage...