We have always been told Bill is just a nickname for William.
Dick is an even crazier one as the nickname doesn't even sound similar to the real name.
Where did the origins of these nicknames start?
There is always more to say, but I answered a very similar question here a couple of weeks back, and you might like to review that response while you wait for fresh replies to your query. Although this answer does not address the William/Bill part of your query directly, I think the comments made about rhyming diminutives do apply across both of your examples:
How did nickname 'Dick' to Richard develop? Did the word dick not always mean penis?