I don't like Billy. William is fine, Willem is nice. But Billy I am not a fan of.
As far as using Billy as a nickname, both Wilhelm and Willem work great. Also, Wilbert, Wilburn, Wilford, Wilfred, Wilkie, Wilton, Willard, Willis, Willoughby (!!), Wilson, Wilmet. Woah, the options are endless.
Three boys nappin', one mama namenerdin'
To peruse my top 100ish names:
http://www.behindthename.com/pnl/82589
"My Inner Birthing Mother Paddles a Canoe" and other non-name musings
found on my blog:
http://jodifriesen.blogspot.ca/2014/07/paddle-power.html