A spokesman for Prime Minister David Cameron said Saturday that he "thinks this is a very sad case and his thoughts are with her family and colleagues."
Throughout the controversy surrounding the hoax, authorities did not identify the nurse. Her identity was released after her death.
Audio of the call posted online suggests a woman spoke briefly to the DJs before she put the call through early Tuesday morning to the ward where the Duchess of Cambridge was being treated for acute morning sickness.
"They were the world's worst accents ever," Greig told listeners Thursday. "We were sure 100 people at least before us would've tried the same thing. ... We were expecting to be hung up on. We didn't even know what to say when we got through."
A tweet from 2Day FM last week after the incident described it as a "hilarious prank."
Off the air, Greig and Christian tweeted about the practical joke on Thursday and earlier Friday, promising "more on the #royalprank." The pair's Twitter accounts were taken down late Friday.
Some listeners applauded the prank, like one who identified himself as Guido on the station's Facebook page and wrote, "It is only a joke people! it was great i love it!!!"
Others were outraged, with negative comments outnumbering positive ones on 2DayFM's Facebook page even before the nurse's death.
"Your stunt was done at a time in this country where there is paranoia about the intrusion of the media into people's lives," Gary Slenders wrote. "I know you will say it is harmless fun, the management of 2DayFM will say that it won't happen again, but this is exactly where the phone hacking scandal started."
The outcry grew exponentially after the hospital confirmed Saldanha's death, leading the Coles supermarket chain to remove all its advertising from 2DayFM.
"This death is on your conscience," reads one Facebook post. Several accused the two DJs of having "blood on your hands."
Saldanha's family released a statement asking for privacy and directing questions to police. She is survived by her husband and two children.
"We as a family are deeply saddened by the loss of our beloved Jacintha," said the statement, released by police.
Saldanha worked at King Edward VII's Hospital for more than four years, and she was described as an "excellent nurse," well-respected by co-workers, the hospital statement said.
The hospital "had been supporting her throughout this difficult time," it said.
A St. James's Palace spokesman said: "The Duke and Duchess of Cambridge are deeply saddened to learn of the death.
"Their Royal Highnesses were looked after so wonderfully well at all times by everybody at King Edward VII Hospital, and their thoughts and prayers are with Jacintha Saldanha's family, friends and colleagues at this very sad time."
Separately, a palace spokesman told CNN: "At no point did the palace complain to the hospital about the incident. On the contrary, we offered our full and heartfelt support to the nurses involved and hospital staff at all times."