By the latter half of the 20th century the increase in commemorative issues and the move towards less stylized designs was putting the convention under pressure. The first minor breach of it occurred in 1967 with an issue celebrating Francis Chichester's solo circumnavigation of the world. It showed Morning Cloud breasting the waves with a tiny blobby figure messing about with her sails who could only be Chichester himself. In 1997 a second breach occurred when the stamp memorializing Freddie Mercury featured another distant blob who had to be Queen's drummer Roger Taylor.
Thereafter the breaches multiplied. A Millennium stamp showed a group of Irish people raising a monolith; their faces were obscured but their friends and family would have known who they were. The Coronation commemoratives in 2003 showed several living people. A stamp marking Britain's winning of the Rugby World Championship in the same year had photographs of the team, shot from the back so no faces were visible, but still easily identifiable.
The big breakthrough came in 2005 when a mini-sheet celebrating the Ashes win showed several England players close-up and full frontal. The Post Office later said this was inadvertent- the product of haste and ignorance-, but I think it's much more likely that someone at the top waved it through thinking, "What the hell; it's a silly rule; lets go for broke."
Since then the living persons have come thick and fast. A set celebrating the RSC gave us a number of living actors, a set of Beatles LP covers gave us Paul and Ringo as well as John and George, a set of images of fantasy heroes and villains gave us Michael Gambon as Dumbledore, Ralph Fiennes as Voldemort and Tilda Swinton as Narnia's White Queen. The Olympics and Paralympics produced a positive blizzard of philatelic portraits of Britain's gold medal winners. And now- bringing us right up to date- we're sticking the 11 faces of Doctor Who (8 of who are still alive) on our envelopes, making David Tennant (who earlier figured on one of the RSC set as Hamlet) the first living person who is not royalty or a gold medal winning athlete to appear on more than one stamp.
This post has been compiled from several sources most especially this article by Peter Jennings (which contains images of most of the stamps I've discussed.)