A statement from the Metropolitan Police said Terry Lineen arrived at Heathrow yesterday, but never made the onward connection to Edinburgh.
He was last seen on CCTV cameras on Southern Perimeter Rd - where passengers are collected - shortly after midnight today.
A Metropolitan Police spokesman at Heathrow told the Herald the man had still not been found.
Sean Lineen has travelled to Heathrow to try and find his father, the Evening Times reported.
His wife, Lynne, said the family was extremely worried and believed the elderly may have been confused after 40 hours in transit and was upset about the anniversary of his wife's death, which is today.
"Whether the emotion of losing his wife plus the long-haul flight has made him disorientated, we don't know, but we don't know where he has gone," she told the Evening Times.
"We are very concerned about him, because he has had his health issues over the years. He has had two hips replaced, and at the beginning of this year he had one of them replaced again.
"He has had bowel and prostate cancer and he has a blood cancer which he is taking medication for. He has all that medication with him, but it might be in his case, so we are also worried about that.
"Obviously, he can't be too long without his medication either."
She said her husband was searching with officers and trying to trace the routes the elderly man may have taken.
It is currently early morning in London, but information was expected to be released over night through the Metropolitan Police.
Terry Lineen's brother, Dennis, said Terry often travelled from his retirement home in Auckland to see his son and daughter-in-law.
He hoped he had realised his mistake and just booked a hotel for the night.
Anyone with information can call police at Heathrow on +44 203 276 1461 or the police non-emergency line on 101
