Lee has set up an initiative, called The Philomena Project, through which she is urging the Irish government to open up more than 50,000 files relating to the estimated 60,000 women who were forced to part with their babies and young children.
It is a race against time because many of the women involved are now elderly and have only a few years left in which to trace their long-lost children.
"I was very upset and unforgiving at the time," Lee said in Rome, a day after meeting the Pope.
"I was hurt and sad and I did lose my religion for a while. But then I got a job as a psychiatric nurse and for 30 years I buried all the pain and pushed it into the background. I dealt with other people's sorrows instead."
Her campaign is backed by the Adoption Rights Alliance, an Irish organisation which has campaigned for more than a decade for the government in Dublin to open up the adoption files.
"We want to know why the Irish government turned a blind eye and why they allowed this trafficking to continue," said the alliance's Susan Lohan. "It paid religious orders to run these mother and baby homes.
"The Irish government carried out the gravest human rights abuses against women like Philomena and their children. We need to open up the dialogue. Philomena's story has acted like a lightning rod for the whole issue.
"The burden of shame and guilt that Irish society placed on her and other women like her should be lifted. Responsibility lies with all of Irish society."
Coogan, who came across the story of Lee by chance in a newspaper article four years ago, also met the Pope and then held a screening of the film for senior Vatican officials. He said he hoped that the Pope would also watch the Bafta-nominated film.
The encounter came on the same day that a United Nations committee castigated the Vatican for continuing to protect paedophile priests who remain at liberty to molest children.
But the comedian and actor said he believed the Pope was intent on tackling the scandals.
"There seems to be a break with the past, from a culture of obfuscation and obstruction," Coogan said.
"They are more open to criticism - they said they didn't see the film as anti-Catholic or an attack on the Church, but as constructive criticism.
"There was an acknowledgment and a desire to engage in dialogue rather than adopt a siege mentality."
The Vatican under Francis was showing "a more open, more nuanced response" to the scandals of the past, he said.
The UN committee was particularly critical of the Catholic Church's treatment of pregnant, unmarried young Irish women who were sent to the so-called "Magdalene Laundries".
The women lived in "slavery-like conditions", were subjected to physical and sexual abuse and were deprived of their identity, the committee said.
"We were given numbers and convent names and not allowed to use our old names," Lee said.
"We had to lose our identity."
Despite all that was done to her, she has recovered her Catholic faith and has forgiven the people who took away her child.
"I don't hold it against anyone, not now anyway.
"How could I go through the last 62 years holding a grudge?"
Coogan, who pointed out that he is not a Catholic, said: "I told the Pope that she is a symbol of forgiveness and reconciliation. She's an ordinary woman who has done an extraordinary thing."
