Cosmetic surgeon Dr. Martin Kelly, 42, was found lying on the doorstep of the family's home. A friend found the doctor and called an ambulance, but paramedics were unable to revive him, according to reports.
He was pronounced dead at the Chelsea and Westminster Hospital, where he also was a consultant.
A spokesman for Scotland Yard told PEOPLE the death was being treated as unexplained and a post-mortem examination is scheduled for Thursday.
McElhone, 36, who also appeared in The Truman Show, is pregnant with the couple's third child. She was filming in Los Angeles at the time of the death and was returning to the U.K. along with their sons Theodore, 8, and Otis, 5
Dr. Kelly's father-in-law, Roy Greenslade, said family, friends and colleagues were in "a terrible state" over the sudden death.
"It's still a mystery. It's so unexpected. Such a fit, handsome, fighting fit young man of 42," he said. "It's just completely baffling. A medical colleague was on hand, so he had the best possible chance but was just unable to revive him."
Dr. Kelly was a leading reconstructive surgeon and the founder of the Facing The World charity, which has treated hundreds of children born with facial disfigurements through pioneering reconstructive operations.