Skripal and Yulia were found unconscious on a park bench on March 4, 2018, after Russian agents smeared Novichok on the door-handle of their home.
They were rushed to hospital and put in induced comas to prevent the poison damaging their organs.
Yulia left hospital in April and was taken by police to a secret location, where she was guarded by British intelligence agents. Her father joined her a month later.
Natalya said: "I am happy to know that they are in New Zealand - if it's true. It's a faraway country. I am hoping they will have a safe start to a new, happier life."
A police officer stands the home in Salisbury of Sergei Skripal on the first anniversary of his poisoning. Photo / Ben Birchall, PA Images via Getty Images
The family said two calls last year by Skripal, 68, to his niece Viktoria - who cares for his 91-year-old mother Elena - feel like permanent farewells.
Natalya, 67, who lives in Siberia and was close to Skripal, said Yulia, 36, went on her family's social media pages in March but did not make contact.
She said: "Maybe they moved in March when Yulia was seen checking out our social media page - I wonder.
"There was a little beacon of hope in March when Yulia visited her own social media and went on to my page.
"'She didn't say anything, but I know she went around all our family profiles. I wonder if this is because she was missing us."
She continued: "I think about them all the time, and pray for their health. I often see Sergei in my dreams. I do hope they find peace and happiness after their ordeal.
"I say to them: Please remember we love you and think of you wherever you are and whatever name you now carry. We are family. You are not alone.'
She added: "I was always very close with Sergei, as we spent so much time together in our younger years.
Yulia Skripal, pictured after recovering from the nerve agent poisoning in Salisbury. Photo / Getty Images
"Back then I always felt a kind of spiritual connection with him, and it looks like this will be the only way I'll be able to communicate with him now."
Viktoria Skripal, niece of former Russian military intelligence officer Sergei Skripal poisoned in Salisbury, UK, speaking in 2018. Photo / Getty Images
He indicated he and Yulia were "fine" but that they had no wish ever to return to Russia.
When she said his elderly mother needed increasing assistance, he replied "here I cannot help you", adding: "Soon we will be taken to another place."
Elena, 38, who lives in the far east of Russian, said she hoped her uncle was doing well.
She said: "We do not know anything about Uncle Sergei and Yulia now. But we really hope that they are doing well.
"After all the hype subsided, I think it is much easier for them to live. I would like to tell them that we love them, ask them to take care of themselves and each other.
"We are happy if they are all right and doing well. I don't know where they are now. They might be in any European country, and perhaps still in Britain."
The report of their new life in New Zealand came a week before the broadcast of a BBC drama telling the stories of people caught up in the Novichok poisoning.
It also followed in February confirmation the pair were alive but no word on where they now lived.
British ambassador to Russia Deborah Bronnert said: "Of course they are alive. I can't tell you where they are because we respect people's right to make their own decisions.
"Both the government and the police will always be guided by wishes of the individuals."
Asked if they were in the UK she said: "Unfortunately, we cannot comment on these matters."
Britain believes GRU agents Anatoly Chepiga and Alexander Mishkin were behind the Novichok poisoning on the Skripals in Salisbury.
