I don't know that you would find a birth certificate in 1812 in England or across the sea. You might be able to locate a baptism. Sometimes parent would wait until they were back "home" to have the child baptised or recorded in the local parish as baptized. Sometimes when that happens the vicar may actually put the birthdate in the record also.
This is the family in 1841?
HO107/317/26/folio 26
Stockton-on-Tees
Paradise Street
Robert Robinson, 30, Butcher, N
Eliza Robinson, 30, N
William Robinson, 10, Y
Mary Robinson, 8, Y
Henry Robinson, 6, Y
Joseph Robinson, 4, Y
Eliza Robinson, 2, Y
Elizabeth Robinson, 4 months, Y
William Kelly, 55, Shoemaker, Y
Thomas Hornsby, 40, Labourer, N
From the 1851 census (HO107/2383 folio 678 page 14) her father William was born in Red Lion.That would be the parish of Hart. You might try a parish marriage in Co. Durham but it did not look good looking at the Joiner data base. Then look for a burial in Stockton for a female Kelly about the same age as William, the record might say wife of .... you will at least get a first name. If Eliza is marrying in 1830 you'd think they would be there at least a year before.
Just some suggestions off the top of my head.
Janis