After unsuccessfully trying to find Ms Miley and inconsistent comments the children had heard over the years, her son Craig finally reported her missing to Dorset Police in December 2019.
Inquiries with the Department for Work and Pensions led to the discovery that Ms Miley was living in London in 2003 but the trail then ended.
Further police investigations found that she had entered the US in 2004 and may be living in Arizona.
The US authorities carried out an address check and in October this year Ms Miley was found to be living there safe and well.
Officers learned that she left Bournemouth in 1991 and went on holiday to Egypt.
She later went to Cyprus and was involved in a road traffic collision that left her in a coma with no recollection of her life before.
Eventually she moved to the US and tried to locate her family, but this was unsuccessful as they had since moved a number of times.
Ms Miley's son Craig Miley said: "Dorset Police have been amazing throughout.
"I came to them with a genuine concern for the welfare of our mother who hadn't been seen or heard from in almost 30 years.
"The team picked up the case and were relentless in their search. No stone was left unturned and they constantly kept me up to date on a regular basis even through a lockdown period.
"The team were extremely professional and had real empathy towards myself and my family.
"I spent many years searching and even tried using a TV crew to find our mother.
"My family and I would like to say thank you to the whole team who helped and wish them all the best for the future."
Mr Miley's sister Christina Summons said: "Words cannot express my gratitude for those involved in trying to locate my mum and reunite us as a family.
"After years of searching I had started to give up hope of ever having my mother in my life so I will always be thankful for the investigative efforts of Dorset Police.
"Growing up as a girl without a mum has been difficult in so many ways, especially after having my own four children.
"They now have a grandma and although the distance between us means we are unable to actually meet, we will not give up hope.
"I have set up a fundraiser in the hope that we can finally close that distance. I look forward to being able to hug my mum, brother and sister and for her to meet her grandchildren.
"So once again, a massive thank you to Dorset Police for giving us hope and a new future."
Detective Inspector Wayne Seymour, of Dorset Police, said: "I am really pleased that through our investigation we were able to reunite a mother with her children after almost 30 years.
"This was a detailed and extensive investigation that involved a lot of multi-agency working and I would like to thank everyone for their efforts.
"We were faced with a number of challenges throughout, but we remained determined and always found a new line of inquiry to investigate, which thankfully led to finding Lee-Tracey."
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