The singer met 13-year-old Shamsher Sherin at a Calais refugee camp last week and was so moved by his story that she apologised "on behalf of my country for what we've put you through".
Shamsher is now with his father in Birmingham, according to translator Mohammad Ishaq, who is helping them.
He was among dozens of child migrants brought to the UK on Wednesday.
They will be joining their families and are among about 100 minors who will be brought over by the Home Office.
More updates on this story and others in Birmingham
During the interview with the pop star, Shamsher revealed he had been in the Calais refugee camp for two months after fleeing his home in Afghanistan.
He said his father lived in Birmingham, giving him a legal right to make his way across the Channel.
But he said it was better to make his way into the country on the back of a lorry because the legal process was too slow.
He described one occasion when he was kicked and slapped by police after a failed attempt to get into the UK.
Speaking to the BBC, he said he looked forward to starting a life in England and wanted to go to school.
After watching the interview, his father Hazrat Gul Sherin, who fled Afghanistan in 2005 and now lives in Alum Rock, said he would hold his son close to his chest and embrace him as soon as they were reunited.
Translator Mr Ishaq said he received a phone call on Thursday evening from Mr Sherin to say he was now with his son and he was "very happy".
"He was so emotional... he wanted to thank the UK government... he said 'my dream is completed now'."
After hearing the boy had crossed the border, Allen tweeted: "So happy to see that Sham Sher made it to the UK safely and won't be risking his life jumping on to moving vehicles."
A refugee whose plight reduced pop star Lily Allen to tears has been reunited with his father after 10 years.