I have the other Henry in 1901, he's a carpenter, Clapham as birthplace, 1861 as birth year, his wife is Emily A born Lambeth, with three children, all born Clapham. Family living in Battersea. In 1901, they are in Clapham, he is a packer in a piano factory, birth year and birthdate as in 1891. In 1911they still can be found in the same part of south London.
So it looks like all the census records for Henry born Islington, with wife born Liverpool, occupation furrier (farrier on one census) consistently b 1859 is yours. So the problem of the wife's name.
Options: Emma changed her name to Emily or Emma died and he married an Emily or he and Emma split up and he is living with Emily as a married couple.
Has Emma Smith been investigated - difficult name, Smith, but with the father's name and occupation should be possible to find her in 1881 when single - was she born in Liverpool?