There are no regular rules for nicknames. I have a friend whose legal name is Andrea. All of her friends call her 'Drea. Some *Andrew*s are called Andy (actually, most of them are, I'd imagine) and not 'Drew. There's no rule that says their nicknames have to drop the "An": they just do sometimes. I've known women named Anita with the nickname 'Nita: only the initial "A" is dropped.
The only rule for nicknames that matters is what the person with the nickname thinks about it. I have a friend who's tall and morbidly obese. He calls himself Tiny Tom.
The famous overweight pool player Minnesota Fats was originally called New York Fats. That was okay by him. Fats Waller (jazz singer and piano player), Fats Domino (R&B singer), and Chubby Checker (pop singer) had no problems with those nicknames.
Nicknames are generally shortenings of given birth names. Turning a multisyllabic name into a monosyllabic one is usually the goal, e.g., Percival often becomes Percy, Elizabeth becomes Liz, or Beth, or Liza, etc., but it could just as easily become Tater or EG (if her name is, say, Elizabeth Gloria), or Goldie if she has blond hair.
The famous French writer Anatole France never did have the nickname 'Atole. He was named Jacques-Anatole-François Thibault and changed François to France, a pen name, not really a nickname.