The family of a British girl who disappeared in Australia more than 50 years ago has threatened to name a key person of interest unless he comes forward to answer their questions.

Authorities believe three-year-old Cheryl Grimmer was abducted from Fairy Meadow beach in Wollongong in January 1970.

A suspect was charged with her abduction and murder, but his 2019 trial collapsed after a detailed confession, made when he was a teenager, was ruled inadmissible. He denies any wrongdoing.

Known only as Mercury, the man’s identity is protected as he was a minor at the time, but a politician has offered to name him under parliamentary privilege as the family pushes for a fresh investigation.

“[Mercury]’s got until Wednesday night,” Cheryl’s brother Ricki Nash told reporters on Friday.

Jeremy Buckingham, a New South Wales parliamentarian who has been supporting the Grimmer family, said he’s prepared to use parliamentary privilege to name the man when state parliament resumes sitting on Thursday.

Mr Nash wants Mercury to explain to the family how he knew information contained in his confession – and if it was true or not.

“Enough is enough,” he said, at times on the verge of tears. “We want the truth to come out.”

Cheryl had been on the beach at Fairy Meadow with her family on 12 January 1970. When the family decided to pack up, Mr Nash, as the eldest brother, had been put in charge of his siblings and told to go to the bathroom block.

Cheryl ran giggling into the ladies’ changing rooms and refused to come out. Too embarrassed to enter himself, Mr Nash went back to the beach to tell his mother to help. When they returned, 90 seconds later, the toddler was gone.

The family had only recently migrated to Australia from Bristol as so-called Ten Pound Poms.

Despite extensive searches, there were no leads. Then in 2017, a man in his 60s was charged with Cheryl’s abduction and murder after officers discovered a confession made to police by a teenage boy in 1971.

A judge later ruled the confession could not be presented as trial evidence and the charges against him were dropped.

On Friday, the family released a lengthy document detailing the missteps they say were made by authorities in NSW in the search for Cheryl, and called for more answers.

“We’ve made various requests to NSW authorities for a fresh prosecution or a fresh inquest but to no avail,” the family said in the letter.

“We feel that we have been fobbed off numerous times by the police, saying that they’re conducting reviews of the case or exploring leads that make no sense to us. The incompetence and negligence in the NSW police investigation of this case over much of the past 55 years is unfathomable.”

NSW Police have defended their conduct, reiterating that homicide detectives are still investigating Cheryl’s disappearance – and that a A$1m reward for information remains on offer.

“Police continue to examine every line of inquiry and search for answers into Cheryl’s death,” NSW Police said in a statement.

Three potential eyewitnesses came forward after the BBC aired the Fairy Meadow true crime podcast in 2022, which has since been downloaded five million times. Their contacts were passed on to investigators.

But the family was recently told that a four-year-long review of the case found there was no new evidence that could lead to a conviction – even though the new potential eyewitnesses were not formally interviewed by officers.

The family, alongside a volunteer team using cadaver detection dogs, also searched an “area of interest” they hoped would be a breakthrough in the case earlier this month. But police said that a subsequent search of the area only found animal bones.

Cheryl’s family disputes the police’s response and said volunteers were back in the area on Friday, collecting soil samples to be sent to the UK and the US for further analysis.

“Cheryl disappeared more than 55 years ago. It’s time for answers, it’s time for accountability,” the family said.

It comes as the NSW parliament announced an inquiry into long-term missing persons cases in the state – including Cheryl’s. It will look into how investigations have been carried out and ways of improvement.