So why would they fight? There is no way Ned would kill his nephew.
He was Robert's right hand man. Sure Ned wouldn't kill him himself, but from what they and even Lyanna knew, he would still turn the child over to Robert. I don't think they had any doubt that Ned would try to protect the child and plead with Robert to spare it, but they knew that that would be his doom. There's a reason Lyanna didn't die until she got a promise from Ned. Ned committed pretty hardcore treason by protecting Jon that could have meant the end of their house if Robert realized it. Lyanna had to secure the promise out of him, and he kept it. No reason the Kingsguard should have thought he would have made such a vow, especially for how loyal he was to Robert.