It was the death notices in the Sheffield Independent that enabled me to connect William Evatt with Henry Royle Evatt.
Death of a brother
I was able to link William back to a Quaker family from Castle Donington because of an announcement of the death of one Samuel Evatt from Castle Donington while staying with his brother William Evatt of Upperthorpe in Sheffield. In fact the three brothers William, Joseph and Samuel all died in that year: 1836
Family
I have not found any records for a marriage or the birth of Henry Royle Evatt. Henry Royle Evatt had his children baptised as Roman Catholics so I do wonder if Henry's mother was a Catholic - in which case the relationship might have been somewhat awkward
Historic Context
Religion
William came from a Quaker family, was recognised as a Quaker in various papers and was buried as a Quaker. His son Henry had his children baptised as Catholics. Quakers were not allowed to marry out until the 1850s. Up until then, if a Quaker married out, he or she was no longer a Quaker.