I think what has been posted is correct even though you are dismissing it:
Children in 1871 are Jane bc.1853, Harriett bc.1858, Anna bc.1865, Julie bc.1867, Amelia bc.1868 and George bc.1870.
The children beginning with Anna all shows in births index and GRO own index confirms their mothers maiden name was Cox:
Anna Maria Mar.1864 Wandsworth
Julia Dec.1865 Wandsworth
Amelia Mary Dec.1867 Wandsworth
George William Dec.1869 Wandsworth
With an 7 year gap between Harriet and Anna it indicates a second marriage for William which again, the GRO births index would seem to confirm:
Eliza Jane b.Dec.1853 Wandsworth
Harriet Sarah Mar.1859 Wandsworth - mmn for them both is Goldsmith!
So, the 2 marriages posted by Mabel Bagshawe, would appear to be the correct ones - with first wife Jane death jun.1862 Wandsworth - GRO index shows age 32 and this ties in with Jane Vass Goldsmith b./bp.1830 in Lambeth.
Looks like William and Jane had a son who died - Samuel William Vickers Mar.1857 mmn Goldsmith, died June 1857 Wandsworth.
All the children's baptisms are on Ancestry - first 3 (with Jane Vass Goldsmith) show father names as William Francis for Eliza Jane (24/11/1853 Lambeth), Francis William for Samuel William (17/5/1857 Battersea) and back to William Francis for Harriet Sarah (13/2/1858 Battersea).
He's William Francis for Anna Maria and Julia but plain William for Amelia Mary and George William.
Annette