I have done place searches of the 1861 census for Conwy. There are two districts and there is no trace of John Williams in District 1.
District 2 has a family living in Chapel St, Conwy. The head is John Williams age 32, married, Cabinet Maker, born Conwy. Also in the household: Ellen Williams, wife, age 28, "Anglesea person"; Ann J Williams, daughter, age 7, born Liverpool; Thomas Williams, age 5, son, born Liverpool; Alice Williams, daughter, age 3, born Conwy, and William Williams, age 1, son, born Conwy.
There is another John Williams living in Mount Pleasant. The household comprises:
John Williams, married, age 35, carter, born Anglesea Aberffraw.
Elizabeth Williams, wife, age 28, born Caerhun.
Hugh Williams, son, age 4, born Dwygyfylchi
Catherine Williams, daughter, age 1, born Dwygyfychi.
Martha Griffiths, niece, age 12, born Holyhead.
Still no trace of Robert!
I have been advised that the marriage certificate for Edward Williams to Lydia Thomas shows his father as John Williams, Labourer. A witness is shown as Elizabeth Williams.
Helig.