I had a very similar experience that might be helpful to you. I will start out by saying though, that I waited until the pertinent party was deceased before sharing what I knew. And I did so very carefully. I provide below only the first names of deceased individuals.
My husband was contacted in 1991 by an older woman, Dorothy, who was seeking to learn about her birth father, Harry, a great uncle of my husband's. She'd been born out of wedlock in 1919 and it wasn't until she was 60 and her mother was on her deathbed (1979) that she learned the name of her birth father, Harry. She found my husband's name on the obituary of HIS mother in 1991. My husband, bless his soul, went to visit Dorothy and her children, and provided her what little information he knew which he had gleaned from HIS father (which wasn't much), only that he knew that Dorothy had a half-brother, Maurice, probably still living.
Fast forward to oh, 1999. I'm doing genealogy and working on my husband's line. He tells me this story and I go researching. I find Maurice, the half-brother of Dorothy. I contact him and interview him, without saying anything about his half-sister, just letting him know I research the family and am contacting living relatives to learn more. He shares with me as much as he knows about his father Harry (who had died in 1973)-- quite a story and he's obviously proud of him. No mention of the woman and child his father had abandoned back in 1919.
From this interview and subsequent research I conducted, I was able to generate a family history report of facts about Harry. I shared this with Dorothy and her family when my husband and I visited them in 2000. She was thrilled to at least know more about him. Dorothy died in 2001 and one of her children attempted to reach Maurice (technically their half-uncle), but he did not respond.
After I learned of Maurice's 2007 death, I drafted a letter to his known children (who would have been half-first-cousins of Dorothy's children). The cover letter explained who I was, that I did family history research and I attached an outline of their shared ancestry with my husband. The outline included their grandfather Harry, and both women by whom he'd had children, and each of his two children, including Dorothy and their father Maurice.
They contacted me by speaker phone-- all three of Maurice's children (now adults with families of their own). They were thrilled and curious. Their father on HIS deathbed had hinted that their grandpa Harry had had another family. Because I knew that Dorothy's children had tried to contact their father and wanted to be in contact, I shared with Maurice's children what I knew and asked them if they wanted to be in contact with their half cousins. Yes! I shared THEIR contact info with Dorothy's children, and then let the half cousins go from there. They ultimately met. And I was able to fill out the full picture of Harry's descendants.
But I did wait until the unwilling party (in this case Maurice, son of the birth-father of Dorothy) had died before making contact with his children. And I shared just documented facts, no "story". That I left to the descendants to share with each other.
