Hi,
I see there is an Edward ROBERTSON who could be Emma's brother,
Edward ROBERTSON  
Baptism: 23 Oct 1836 
Father: Robert ROBERTSON,  Confectioner 
Mother: Mary Ann 
Residence:  Brompton Row
Parish or Poor Law Union: Brompton Holy Trinity 
Borough: Kensington and Chelsea 
I found these two in 1841,
HO107/688/16  f.7  p.8   Chelsea
Royal Hospital Chelsea
/
Elizabeth WILTSHIRE    45    School Assistant               N
Emma COOPER            15    Daughter of master baker    N
Ashly      "                  9    Son of    "                         Y
Emma ROBERTSON       5    Step Daur                          Y
Edward ROBERTSON     4       "    Son                          Y
/
A very odd household as the COOPER children seem unrelated to Elizabeth WILTSHIRE. I also can't find any WILTSHIRE/ROBERTSON marriage which would fit.
Regards,
Daisy