Green’s Dictionary of Slang

(US) to act in a respectable manner, often excessively and interferingly so.

1954		Collans & Sterling House Detective 127: The lecherous couple ... will be ‘surprised’ in flagrante delicto, as the legal bigwigs nice-nelly it [HDAS].
2000		G. De Mare Sybille n.p.: Ah been livin’ lak a man foah seventy-six yeahs an’ ah ain’t goin’ to nice nelly foah any o’ you creep.