I have come across several William Rushton's in the 1881 and 1901 British censuses. However, I simply do not know whether a William was a son of my great grandfather John. He was married three times. My grandfather, Peter, was the eldest son by the first marriage. I know of a sister, Lizzie (Elizabeth) and a step sister, Cissie (Cecilia I think) by the second marriage. My grandfather did not get on with his stepmothers and my late mother implied that they did not see much of grandfather John. So later siblings are not known to us. Your grandfather William would be the right age for a son, but I have no knowledge yet of any confirmed relations.
Have you tried the Mormons site and the British 1901 Census site? They are both very useful.