We also hear about Wylla from Godric Borrell, the Lord of Sisterton. There’s no explanation for that save Ned Stark spreading the story to make sure that people have a cover story for Jon Snow.
We do not hear about Wylla from Borrell. No idea where you got that he named her. Also, your sequence of events is very off. You seem to think that Ned spread Wylla’s name and the rumor around the Sisters when he came home
from the Rebellion. But the story Borrell tells is about the beginning of the Rebellion, and is not about Wylla the wetnurse at all.
“Ned Stark was here?”
“At the dawn of Robert’s Rebellion. The Mad King had sent to the Eyrie for Stark’s head, but Jon Arryn sent him back defiance. Gulltown stayed loyal to the throne, though. To get home and call his banners, Stark had to cross the mountains to the Fingers and find a fisherman to carry him across the Bite. A storm caught them on the way. The fisherman drowned, but his daughter got Stark to the Sisters before the boat went down. They say he left her with a bag of silver and a bastard in her belly. Jon Snow, she named him, after Arryn.“
The only Wylla in ADWD is Wylla Manderly. And it’s far more likely that the gossipping Sistermen wanted to claim honorable Ned Stark’s famous bastard for their own than any rumor spread by Ned. (As it would have been about a year between the time Ned was in the Sisters and when he came home with Jon Snow.)