Australian Melanie Greig posed as the Queen during the call in which private medical information about the pregnant Kate was disclosed in December 2012.
Jacintha Saldanha, 46, who took the call, hanged herself three days later as the call made headlines around the world.
Addressing the mother-of-two’s family after an inquest today ruled her death was a suicide, Ms Greig sobbed as she told them and the packed courtroom: “I really just wanted to say I am truly sorry, I’ve wanted to say that for so long.
“This tragedy will always stay with me and serve as a constant reminder.
“To the second nurse involved, I am so deeply sorry for what you have had to endure. I pray you have found the strength to live on as best you can.
“I was always concerned about the wellbeing of both nurses and I wish I’d tried harder to stop that prank from being aired.”
She went on to urge hospitals and the media to learn from the incident and make sure it was not repeated.
She added: “To fellow announcers and DJs, I urge you to speak up if you don’t feel comfortable and consider the feeling of others when trying to make a joke.
“The joke should always be on us, the DJs.”
Indian-born Mrs Saldanha was found dead in nursing accommodation on December 7 2012, after Greig and fellow shock-jock Michael Christian broadcast the call on Sydney’s 2Day FM on December 4.
Saldanha’s only role was to answer the phone to the DJs and, believing the call was from the Queen, put them through to a nurse who revealed details of Kate’s condition.
The royal was being treated at the King Edward VII’s Hospital in London for hyperemesis gravidarum, a form of severe morning sickness.
But the two-day inquest at the High Court heard that she held herself responsible for the mistake, despite the private hospital’s management supporting her and the other nurse as victims of a cruel joke.
Management described her as robust and resilient professional who gave no indication she was not coping with the furore.
Leicester MP Keith Vaz, speaking outside on behalf of Mrs Saldanha’s family, said: “Over the last two days, the family and the world have heard about her nobility, her dedication to her patients, her love of her job and her fantastic professionalism.
“The family of course accept the verdict of the coroner’s court today.
“It is an irony that four calls made in 115 seconds, which were the cause of so much mirth in Australia, could have deprived Benedict (Barboza, her widower) of his wife and Lisha and Junal of their beloved mother.
“These despicable and cruel actions, this hoax, has changed their lives forever.”
He added that the family was taking advice over possible future legal action.
Junal thanked friends and family in their home city of Bristol and “everyone in Britain and around the world who has supported us through this difficult time in our lives”.
He said: “My dad, my sister and myself miss our beloved mum every day and will do so every day for the rest of our lives.”
The nurse who disclosed Kate’s condition yesterday told the inquest that Mrs Saldanha emailed her after the prank, saying she was “terribly sorry”, and added: “It’s all my fault. I feel very bad about this to get you involved. If there was anything I could do to mend this I would do it.”
The nurse, who cannot be identified for legal reasons, said Mrs Saldanha had given her the impression she was talking to the Queen.
Another email the panicked nurse sent to a second colleague about her “shame” after the hoax made international headlines was read to the inquest today.
In it she said: “It’s all my fault. I don’t know how to face my bosses tomorrow. I feel so ashamed of myself.”
The inquest heard that in the hours before her death she had read news reports of the hoax call online.
Westminster coroner Dr Fiona Wilcox said there was evidence the hoax had been “pressing on the mind” of the nurse before she killed herself, along with her difficult relationship with a junior colleague who had made a complaint of bullying and harassment against her, which had recently been dismissed.
She did not criticise the DJs or the radio station but, addressing four calls made to the hospital by production staff to gain Mrs Saldanha’s consent before the recording aired, added: “If she did take these calls I find it inconceivable she would have consented, as a participant in the call, to its broadcast.”
Andrew Robertson, current chief executive at King Edward VII’s Hospital, described her as a “first-class nurse”.
He added: “The coroner has thoroughly investigated what happened and the evidence has shown that King Edward VII’s Hospital did everything it reasonably could to support Jacintha following the hoax call.”