Wylla is a servant of House Dayne, who served as Edric Dayne’s wet-nurse, and who Ned claimed as Jon Snow’s mother when asked by Robert Baratheon.
My belief is that Wylla was both a wet-nurse and a midwife and had been sent by Ashara Dayne to the Tower of Joy to help her brother Ser Arthur prepare for the impending birth of Rhaegar’s child.
As a result, she was one of the few people who was there for Jon Snow’s birth. We know that there was someone else that Howland Reed who was there at the Tower of Joy since “they had found him still holding her body, silent with grief. The little crannogman, Howland Reed, had taken her hand from his.” The plural there means that there had to be someone there at the ToJ who didn’t partake in the fight, and her being busy at the birthing would explain that.
She then went with Eddard to Starfall and was Jon’s wet-nurse for the time that Ned spent at House Dayne’s seat. I also think that she may have accompanied Eddard briefly to King’s Landing (Jon probably still wasn’t on solid foods), because Robert remembers her.