"I expect we will get a lot of rubber neckers coming past but I'm hoping it won't last long," he said.
Mr Jones and his wife, Caryl, had been in regular contact with Lundy since he was convicted of the murder of his wife, Christine, and daughter, Amber, in August 2000.
"We've always seen him as often as we could ... It's become more regular since he was transferred to Rangipo [Prison, near Turangi]."
The couple attended the three-day Privy Council appeal hearing held in London in June.
On Monday, the Privy Council released a reserved decision allowing the appeal, quashing the murder convictions and granting a retrial.
Mr and Mrs Jones visited Lundy at Rangipo Prison, where they have visited him regularly on Sundays, as soon as the decision was reached.
Mr Jones said Lundy would also phone them two or three times a week from the prison.
He was "ecstatic" when the couple saw him on Monday. "He was very happy ... quite relaxed. It's a huge weight off his shoulders.
"Finally the facts have been listened to after all these years."
Mr Jones said having Chief Justice Dame Sian Elias on the five-member Privy Council panel had been crucial.
"She is a New Zealander and she listened to the facts. The appeal decision was unanimous, five to zero."
Mr Jones said he had "no idea" who might have committed the murders.
"I honestly don't think the police adequately investigated all of the case ... They stuck to the 7pm to 7.30pm [time-of-death] timeframe."
Police now had a "huge task" ahead to reinvestigate the murders, he said.
"There were around 60 suspects who had alibis which police will have to go back and recheck."