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In the absence of a time machine, these are guesses.

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Margaret: shortens to Maggie very easily. If you're bored or playful, Maggie can morph into Meggie. Once that's done, all you need is a sibling with a bad cold (this is the UK we're talking about, back in the Middle Ages probably) and you get Peggy.
Molly and Mary: I also detect the presence of a sibling here. R sounds are notoriously difficult for children, so Molly could be their best attempt at Mary. And it took on a life of its own and, same as Meggie, morphed into Polly.
Jack, or Jock in Scotland, = Johnkin = little John, like a lambkin! And it got shortened.
Bob and Bill also seem to be likely mistakes made by the very young; babies and toddlers find /b/, /d/, /p/ and /m/ easy to say (think baba, dada, papa, mama ...). Parents think this is cute, and perpetuate it.