I've met a few babies called Billy lately. I like it, but I would definitely give him the full name William so that he can decide for himself when he is older. He might feel William is more appropriate when chairing a board meeting.
I know 2 children called Billy, a 9 year old and a new baby. Great name. 9 year old never had a problem. He is unbelievably cool with surfer curly blonde hair and attitude though. Which could explain it!