Yaanis Mellersh, eight, and Mira, six, and their German mother, Elke Mellersh, 45, were found dead in Soke on Friday.
A German court had granted custody of the children to their father, Justin.
It is feared the Turkish authorities will give custody of the three bodies to Elke's relatives, meaning the children could be buried in Turkey.
Natasha Mellersh, Yaanis's godmother and cousin of both children, told the BBC: "Hearing the pain in my grandfather's voice just made me completely devastated.
"I later spoke to my mother, we just cried, it was just heartbreaking.
"Now the main thing is to get the bodies back into the country and bury the children. After that we can really grieve."'Inner pain'
Justin Mellersh, 46, and his father Nick are in Turkey and were due to go to the mortuary to see the children's bodies on Tuesday.
In a message posted on Facebook, he said: "Yaani and Mimi - the lively, loving, tender, kind individuals who were children, grandchildren, nephew, niece and cousins to us - lost their lives because of deliberate actions of their mother, who took her own life as she took theirs.
"We do not blame the mother for this tragedy - it's clear that she suffered a debilitating mental illness with paranoid delusions.
"Rather we mourn that we were not able to ease her inner pain and confusion, and so were unable to prevent this sad, sad end."
Mr Mellersh's family live near Lyndhurst in the New Forest.
The two children grew up in the area before moving to Germany with their parents.
They were reported missing in 2010 and their mother was wanted on suspicion of child abduction in Germany.'Failing the children'
The bodies of the children and their mother were found in a farmhouse in Soke.
They were last seen at Hepstedt near Bremen in Germany in February 2010.
The children's aunt, Lucy Mellersh, wants to see action taken to protect and recover children who have been victims of parental child abduction.
She said: "Every well-intentioned person who helps to hide an abducting parent is failing the children and putting them in danger.
"Every slow-moving government agency that assumes the children will be okay because they are with their mum is putting the children in danger.
"Every police officer who puts a parental abduction to the bottom of the pile on their desk is putting the children in danger.
"Don't leave it until it is too late, don't focus on last Friday's deaths. Do something about the children who are still missing."