Do you have access to Workhouse records?
The Wandsworth workhouse birth is on Ancestry, in their
London Births and Baptisms database. Many children born in a workhouse/infirmary were still unnamed when the birth register was written up, so it can be helpful to search by mother's name rather than child's.
Unfortunately, for the Wandsworth & Clapham Union, there are almost no other surviving records before about 1870.
What's really interesting is that the family lived in Clarendon Street, Paddington in later years.
You'll find all the Paddington addresses are nearby to each other.