All three were repeatedly stabbed and left for dead within a year of Miss Nickell, who was knifed 49 times by Napper in front of her son Alex as they walked in Wimbledon Common in 1992.
Despite arresting several suspects for the unsolved murders police have never found who was responsible.
Officers will now visit Napper at Broadmoor where he has been detained indefinitely. It is likely they will question him about the deaths of Claire Tiltman, 16, Penny Bell, 43, and 47-year-old Jean Bradley.
Mrs Bell's daughter Lauren Bell was just nine years-old at the time and has spent most of her life longing to see justice for her mother.
She was told yesterday (Friday) by police that they are re-opening the case and intend to question rapist killer Napper.
Mrs Bell, a businesswoman, was stabbed to death as she sat in her car near a leisure centre in Greenford, west London in December 1991.
Miss Bell said she was ecstatic that her mother's killer might finally be uncovered.
The 26 year-old, who is a customer services manager for a car manufacturer, said: "I was absolutely thrilled to see justice finally done for Rachel Nickell, so that her family can have closure on what has been a terrible time for them all.
"I have never had that closure. To solve the mystery of my mum's death and have the right person put behind bars would be a dream come true for me.
"The police did not give me any idea of the timeframe and made no promises, but I'm thrilled that at last there is some movement on my mum's case after all these years. It has been 17 years of not knowing, and still to this day there are details about my mother's death that I have not been told.
"All I have ever wanted is to know the truth, but not knowing makes life awful.
"It completely freaks me out that the beast who killed my mum could still be out there."
Speaking from her flat in the village of Tylers Green near High Wycombe, Miss Bell said she had always wondered whether the same killer was responsible for her mother and Miss Nickell's death.
"As a family we have always followed the Nickell case because it was so similar to my mother's case, in terms of the timeframe and the brutality of both murders," she said. "I don't have the expertise to say whether I believe Napper is the man responsible for my mum's death.
"However, due to the similarities of the two cases it has always been at the back of my mind that he could be the killer.
"I'm delighted that the police are going to speak to him about it."
Claire Tiltman was stabbed more than 40 times in Kent six months after Miss Nickell, while Miss Bradley was stabbed 30 times in west London two months later.
Claire's father Cliff has asked Kent Police about potential links to Napper in light of his conviction this week.
"They say they have nothing conclusive so cannot comment," he said.
"I can't condemn Robert Napper because I don't know if he did it or not.
"Over the years there have been a number of people put in the frame for my daughter's murder."
Napper has been at Broadmoor since 1995 after admitting killing 27-year-old Samantha Bisset and her four-year-old daughter Jazmine the year before.
It is feared he could be linked to as many as 106 rapes and sexual offences on 86 victims.
Laurence Alison, professor of forensic psychology at Liverpool University, said the three unsolved murders were striking in their similarity and should be revisited.
"Statistically, frenzied knife attacks that appear motiveless are pretty rare," he said. "These were all in the early 90s, the victims suffered multiple knife wounds and certainly two of them were geographically close.
"They may or may not be linked (to Napper), but it would be important to carefully consider them. Improvements in technology and analytical procedures mean police can explore these links systematically - an option that wasn't open to them 15 years ago."