Christine Lundy's brother, Glenn Weggery, said news of the appeal was "very tough" for the family.
"It's 12 years later, we're trying to move on with our life and let Christine and Amber rest in peace and it's just not being allowed to be done," Mr Weggery told Radio New Zealand.
"It keeps getting dragged up every few months.
"He's got the right to appeal, fine. But it's been dragged out for so long, and it took so long for them to lodge their appeal when they've been talking about it for years.
"Did any of them ever care about Christine and Amber? Because they sure as hell don't care about their families."
Mr Weggery said he was angry to have heard the news from media outlets.
"I'm rather p***** off that victims in this country are so badly thought of that no one has the balls to inform us before the media gets hold of these things," he said.
"Even [Lundy's] supporters could have had the balls to call us. They know where we are. They don't think about the victims."
Mr Weggery said the family would like to travel to London for the hearing but they could not afford to.
Lundy's London-based lawyer, David Hislop QC, said the appeal would pivot on the science used to identify brain tissue found on a shirt.
"We say [it was] flawed science, bad science, and we obviously want to argue that," Mr Hislop told RadioLIVE this morning.
"It was never good science. In essence, what was deployed from the scientists from Texas was a scientific experiment. He'd never done it before, the science world had never done it before and we say he's got it wrong."
Mr Hislop said the argument about whether Lundy had time to commit the murders by driving from Wellington to Palmerston North and back in two hours and 58 minutes in peak hour traffic had "very little to do" with the appeal.
Mr Hislop said he was "delighted" at the Privy Council's decision and he had not yet spoken with Lundy or lawyers in New Zealand.
"We're delighted. We've put a lot of hard work into this, and we want to see that Mark gets the very best opportunity that he can to put his story across," he told RadioLIVE.
Mr Hislop said the defence team would need to raise money to fly New Zealand scientists to London to give evidence as part of the Privy Council appeal.
Lundy's defence relied on evidence from those scientists, he said.
If the appeal was successful, the case would likely be sent back to New Zealand for a retrial, Mr Hislop said.
The Privy Council spokesman said the right to appeal was granted by three justices of the court who decided on submissions by Lundy and the Crown whether there was merit for an appeal to take place.
"Given the high profile of this case, we will also seek to make arrangements to live stream the hearing from London over the internet," the spokesman said.
New Zealand Police found out about the Privy Council's decision through the media this morning, a spokesman said.
He said police would meet Crown Law to consider the appeal.
Police would not comment on criticisms by Mr Hislop that detectives had relied on unrecognised science to charge Lundy with the murders.
The Texan scientist whose evidence about the brain tissue was crucial in Lundy's arrest told APNZ he had been called as a witness for Lundy's Privy Council appeal.
Dr Rodney Miller, the director of immunohistochemistry at ProPath in the United States, said he would not comment on claims his evidence was based on "bad science".
He has previously said his evidence at the trail was "the most important piece of evidence pointing toward Mark Lundy's guilt".
In a paper written in 2003 about his evidence, Dr Miller said he was sought out by Detective Sergeant Ross Grantham in mid-January 2001 regarding the Lundy murders.
Dr Miller said Mr Grantham had already contacted forensic laboratories in New Zealand, Great Britain and the United States and been told they could not help in identifying whether Christine Lundy's DNA found on her husband's shirt was brain tissue.
"Despite my inability to guarantee him success, he decided to bring the evidence from New Zealand to ProPath to see if we could prove that there was brain tissue on Mark Lundy's shirt," Dr Miller wrote.
He was later flown to New Zealand for the trial and testified that the DNA on Lundy's shirt was his wife's brain tissue.
Dr Miller wrote "the definitive identification of brain tissue on Mark Lundy's shirt using immunohistochemistry was the most important piece of evidence pointing toward Mark Lundy's guilt".
Dr Miller, when contacted by APNZ today, said he had been advised by his lawyers not to comment on Lundy's appeal.
