Prince William is called Wills by his family and friends, apparently. It seems it's always been like that, and I suppose being a Prince of the House of Windsor it wouldn't be acceptable to call him Bill, or just plain Will - it has to be Wills! But at least his brother Henry is called Harry, the usual abbreviated form for Henry. Although previous Royal Henries on the throne of England (is Henries the proper plural form of Henry?) have been called Hal for short, eg the battle loving Henry V, the hero of Agincourt, who fell for the daughter of the French Dauphin by the name of Katherine - he called her Kate and quite right too!
As for his grannie - Queen Elizabeth II - she is, and always has been, called Lilibet by all her family and friends - of her own generation I would guess. The name stuck simply because, as a very wee girl, the nearest she couild get to pronouncing the name Elizabeth was just that - Lilibet. So a Lilibet is what she became - in her close family circle. Bet or Betty was out of the question, I reckon. Loads of people call her Liz though - very unofficially.
As for his grannie - Queen Elizabeth II - she is, and always has been, called Lilibet by all her family and friends - of her own generation I would guess. The name stuck simply because, as a very wee girl, the nearest she couild get to pronouncing the name Elizabeth was just that - Lilibet. So a Lilibet is what she became - in her close family circle. Bet or Betty was out of the question, I reckon. Loads of people call her Liz though - very unofficially.