Hi and welcome to Rootschat.
I have had a look at the 1871 census and have found this family which I feel must be them! Although Josiah is listed as Joseph, but wife is correct and so are children, although again Josiah is Joseph, it was not uncommon to get names wrong on census though.
1871 census
address is hard to read but look likes 18 Dawsons Road
Joseph Smith, head, age 35, general dealer, born Kent Woolwich
Alice, wife, age 35, basket maker born Berkshire??? (cant read)
Frederick, son, age 12 born Surrey Wandsworth
Joseph, son, age 10, born Surrey Streatham
William,son, age 4 born Surrey Croydon
John, son, age 1 born Surrey Croydon
Edward Cooper, visitor, age 42, basket maker born Kent Plumstead
Henrietta Cooper,visitor, age 60, basket maker born Kent Plumstead
Frederick Lane, visitor, age 26, ?? worker born Berkshire Reading
Alice Smith, visitor, age 12, born Surrey
RG10 846
folio 53 page 29
Croydon, Surrey
I will see what else I can find.
Nicky