William does not automatically become Prince of Wales. That title merges into the Crown upon the accession of a monarch and no longer exists. William and Kate, previously the Duke and Duchess of Cambridge, which were created by the Queen in 2011 became the Duke and Dutchess of Cornwall, the peerage title for the heir apparent.Yes, we tend to forget Kate's title will change.
Elizabeth II gave Charles the title Prince of Wales in 1958 by Letters Patent. That title is not automatic although King Charles may choose to bestow in.
Right now, William is a prince, as son of a monarch, although NOT Prince of Wales (a title that no longer exists as of today) and Duke of Cornwall. Kate is not a princess I don't think (yet) but she is Duchess of Cornwall and is entitled to the style HRH.
I admit, I am not an expert and this is very confusing! But that is what I have been able to piece together. Maybe a resident Royalist will correct me.