Both men were previously falsely accused of historical sex offences.
Both men were falsely accused of historical sex offences and joined forces with pressure group Falsely Accused Individuals for Reform (Fair) to campaign for changes to legislation.
Three years ago the pair said they wanted to “redress the balance” in the legal system as they launched a petition to see those accused of sexual offences remain anonymous until charged, saying this was needed to “protect the reputations of all innocent suspects, whether well-known or not, from the lasting stigma of a false sexual allegation”.
The group said the petition received 27,000 signatures before it “had to be abandoned” when the general election was called.
Now they are pressing the Government to include an amendment in the next criminal justice bill because they believe the “law on privacy provides inadequate protection”.
At a press conference in the House of Lords on Wednesday afternoon, the pair were joined by former Tory MP Harvey Proctor and barrister Hugh Tomlinson QC to make their case.
The group want the amendment to make it an offence in England and Wales for someone to identify or publish information about another person being subject of an investigation “in respect of the alleged commission of a sexual offence” unless charged or if there is a court order permitting this.
Sir Cliff, 81, won his privacy case against the BBC over its coverage of a South Yorkshire Police raid on his home in Sunningdale, Berkshire, in August 2014, following a child sex assault allegation.
The veteran star denied the allegations, he was never arrested and in June 2016 prosecutors announced that he would face no charges.
He told the audience of reporters and campaign supporters he has learnt “how desperate it is to be accused of something you never did. We just need to change it (the law) a little bit. It’s just a compromise”.
He added: “Every single one of us is innocent until proven guilty, so all we are asking is… that you are not named until you are charged.”
Sir Cliff branded the internet as a “disaster area for most people now”, claiming his name will “forever” be on the dark web “as the man accused of the dastardly deed”, adding: “You can’t trust anybody anymore.”
Being falsely accused could “completely destroy you”, he said, adding that at points he would awake with his pulses racing and feared he was “going to die of a heart attack”.
“I can’t express it strongly enough to know what it’s like to be an innocent man and also know that the person that accused you has anonymity in perpetuity.
“I’m past that terrible time but will I ever get over it? The answer is no,” he added.
Mr Gambaccini – who was arrested in October 2013 over claims he sexually assaulted two teenage boys as part of Operation Yewtree set up in the wake of the Jimmy Savile scandal – called on parliamentarians to help “restore the reputation of this country as the most just nation on earth”.
The 73-year-old, a regular fixture on the airwaves for decades, spent a year on bail before the case was dropped.
He said he believed the UK was the “most humane country in the world until the events of the last decade proved otherwise”.
Since then he and Sir Cliff had become “a magnet for people who either are being wronged, or who feel they are being wronged or claim they are being wronged by false accusation”, Mr Gambaccini said.
Addressing concerns from opponents to the plan that anonymity before charge would hinder victims coming forward and police trying to build a strong case, he said: “If the police are professional, they can take the case they have been given … and they can judge it on its merits. They don’t need bandwagoners to do that.”
If the accused is believed by the courts to be a threat in the “present moment” they could be named, Mr Gambaccini added.