The Australian dust-storm is Willy Willy. Willy Nilly, i.e. whether you want to or not, comes from the Latin Volens Nolens, or grammatically Nolens Volens. Pronounce the V as a W.
Willy nilly is a corrupted form of 'will he, nill he'. 'Nill' is an obsolete word being the negative of 'will' - roughly meaning 'won't'. So 'willy nilly' means ' whether he will or won't'.