Hi Rose
The certificate of marriage to George Noyce should show the name and occupation of Annie's father and this information may make it easier to find the family in 1851.
Meanwhile, Valda's comment about her age reminded me that one of my gGrandmothers subtracted ten years from her age to marry an under-age boy, who added one year to his age. Thinking along these lines I have found an Ann Smith, age 13 in 1851, born in Kensington, with parents, Richard, 48, Lab gen, born in Huntingdon, and Elizabeth, 46, born in Alton, Hants. There is also a younger girl, Mary, 11, also born in Kensington. This family are living at 6 William Place, Kensington. (HO 107 1468 407 p53).
There is an even older Ann Sarah Smith, 15, born in Westminster, and living there with parents, Richard and Elizabeth. Richard is a servant, age 50, born in Westminster. (HO 107 1480 414 p10).
These are the only two of 87 Ann* Smith, with father Richard, born between 1835 and 1845, that I can find. There are two others born in Middlesex and one or two in Surrey.
All the best
Gobbo