See thread 45584, (......Carter)linked above.
The "Nobody but the baby" line is absent in the versions collected by Scarborough. It is said to appear in the version collected by Mrs. Sidney Carter, and it was picked up by Gillian Welch, who made up her own variant.
One of those oddities which probably has no significance. The version by Gillian Welch may be based on trad, but is not trad in itself.