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Lucy's a name in its own right, though often seen as a nickname bcause of Lucille Ball or Lucinda!
Jack is so popular in the UK these days - if I wanted a Jack I'd give him the formal registration John. I'd never give a child a nickname, even if I intended to use only the nickname. I think they should have the complete formal name to fall back on. What's cute for a child may seem less so when they're grown! Also, the name's meaning is tied up with its longer form.
The exception might be Annie, which is not so very far from the root word, and some of the traditional spinoffs of Elizabeth.