I hoped the Harrisons/Muirs in St Helens would have been in Keswick or Whitby by 1861 but unfortunately not, they don't seem to have moved about much after that.
I think this is the most likely William J in 1901
RG13/3970 Fylde Blackpool Folio 44 Page 44
113 Handsworth Road, Blackpool
William J Harrison Head M 40 Builders Clerk CC Worker Lancashire St Helens
Mary E Wife M 28 Staffs Burton on Trent
Clifford Son 1 Lancashire Blackpool
Death William Harrison Age 71 December qtr 1894 Prescot Lancashire 8b 457
Death Ann Elizabeth Harrison Age 74 June qtr 1900 Fylde Lancashire 8e 528
RG11/3734 Folio 33 Page 4 Prescot St Helens
1881 1 Cotham Street Windle St Helens
William Harrison Head M 57 Builder employing 2 men and 15 boys Cumberland Keswick
Ann E Wife M 55 “ Wife Yorkshire Whitby
Annie J daur Unm 22 Lancashire St Helens
William J Harrison Son Unm 21 Builder Son “ “
RG11/3651 - Folio 113 - Pg 61: 20 Linnet Lane, Toxteth Park, Lancashire
Mary Barrow Muir abt 1862 London, Middlesex, England Daughter
James E. Muir abt 1866 Sri Lanka Son
Charlotte G. Muir abt 1868 At Sea Daughter
Gertrude Muir abt 1873 Liverpool Daughter
Reginald L. Muir abt 1871 Bombay, India Son
Emma Harrison abt 1857 St Helens, Lancashire, England Visitor
Mary Gilchrist abt 1836 Lanarkshire, Scotland Visitor
Robt. Gilchrist abt 1829 Montrose, Scotland Visitor
Ann Grant abt 1857 Liverpool Servant
Remember this Rambler? Emma Harrison is William and Ann Elizabeth’s daughter.
1871 RG10/3863 Folio 175 Page 36
West Balderson Street, Windle
Wm Harrison Head M 47 Builder and Contractor employing 79 men and 10 appr Keswick Cumb
Ann E Wife 45 Wife Whitby Yorks
Emma M daur 14 Scholar St Helens Lancs
Annie J “ 12 “ “ “
William J Son 10 “ “ “
Mary Muir Mother in Law Widow 66 Retired Whitby York
1861
81 Duke Street, St Helens
William Harrison Head M 36 Joiner and Carpenter Manager Cumberland Keswick
Ann E Wife M 35 Yorkshire Whitby
Emma M Daur 4 Scholar Lancashire St Helens
Annie J “ 2 “ “ “
William J Son 1 month “ “
Ellen Watkinson Servant Unm 19 House Servant Lancashire Rainford