The first time 'silly billy' was used in a general way to refer to anyone silly - rather than someone silly but specifically called 'Billy' - was in the 1870s in a publication by B Jerrold.
More recently, it was brought back into fashion by Dennis Healey, the Labour Minister during the 1970s.
No, it was not brought back into fashion by Denis Healey. It was popularised by the comedian Mike Yarwood doing impressions of Denis Healey. Healey never used the phrase himself, except when he was doing impressions of Mike Yarwood doing impressions of him.
Yes, I, 'Silly Billy' was originally the nickname given to William Frederick, Duke of Gloucester (1776 - 1834) and also to King William IV (1765 - 1837), as you say.
Point taken, Bernardo. The picture I had in my head was, indeed, of Mr Healey caricaturing his Yarwood-inspired self.