The An Education star was initially cast as the young female lead, Rose, in the picture, but dropped out to concentrate on Wall Street: Money Never Sleeps.
Rising star Andrea Riseborough replaced Mulligan, but investors feared the film would not be able to pull in cinema audiences without a big name attached, and they threatened to drop out of funding the production.
Director Rowan Joffe set out to cast a famous face - and he was relieved when Mirren stepped in to play the older female lead, Ida.
He tells Britain's Telegraph magazine, "(Mulligan) walked off to go and do Wall Street: Money Never Sleeps, which is a perfectly respectable decision. She had to do what she thought was right. But that did cause us some quite serious problems.
"(Mulligan's departure) seriously damaged the investors' confidence in the project. (Distribution company) Studio Canal and (co-producers) Optimum Releasing were very supportive, but they basically said, 'The movie isn't happening unless you land us a name'. It was teetering on the brink.
"(I) went hell for leather in pursuit of Helen Mirren, whom I'd always imagined as an Ida, but who hadn't been available to play the role initially. And thank goodness she was available to shoot the movie. Because that, in a sense, did greenlight us."