Hi Al in Vane,
The family do appear to be pretty elusive in 1861
The 1881 census info. posted was from IGI familysearch, perhaps the birth place of William J WILKINSON has been mis transcribed. If that is the correct family I'm wondering if Josiah (William) WILKINSON & Elizabeth WILLIAMS married in St. Woolas, Newport in 1860 as some of the children were christened St. Woollos, Newport.
Perhaps you may be able to find help with their marriage details from this link, couldn’t see a marriage in Trevethin Parish but did find a christening for Elizabeth WILLIAMS father William mother Margaret
http://freepages.genealogy.rootsweb.ancestry.com/~monfamilies/stwoolosnptprt.htmIGI
extracted christenings Trevethin
parents named William & Margaret
Elizabeth WILLIAMS 13/11/1836 Rachel WILLIAMS 16/6/1822
John 18/11/1825
William 22/1/1826, 20/11/1826 & 16/5/1840
Abraham 25/4/1828 & 21/3/1829
Henry 13/11/1836
http://freepages.genealogy.rootsweb.ancestry.com/~monfamilies/trevethinprt.htmTrevethin Parish Records - could see these two children's names
18/11/1825 John Williams s William Williams Margaret Williams Sowhill, miner,
21/03/1829 Abraham Williams, s William Williams Margaret Williams Sowhill, miner
You may already have these details
Could be your Josiah
Josiah
WILKINSONch 16/6/1841 Nuneaton
Father William mother Ellen
Robert WILKINSON 24/3/1844
Ellen 17/1/1847
James 13/5/1849
Harriet 5/9/1852
Marriage13/5/1839 St. Michael, Coventry, Warwick
William WILKINSON single father William
Ellen RHODES widowed
Perhaps zekestone will be able to offer more info.
Good Luck with the search