This question refers to people in a previously posted question about the identity of the mother of Elizabeth Cowan, b. 29 Sep 1824, England, and d. 14 Oct 1896, Ohio, USA: Could "Our Mother" be a term of endearment on 1855 headstone?
Elizabeth's death record in Ohio gives her mother's name as Elizabeth, father's name is Robert (see older question). A Harriet Cowan age 72, is living with Elizabeth and her husband in Ohio in the 1850 census, and her headstone from 1855 reads "Our Mother" (see older question). Notes from an older relative give the father's name as Richard, although that is not based on any records, as far as I can tell.
In a biography of her husband in The Centennial Biographical History of Columbus, Ohio (1901) https://archive.org/details/centennialbiogra00lewipu/page/668: “She was born opposite Hyde Park, and came to America when about 18 years of age.” This was likely informed by her husband, who was living at the time. It also has "Her father, Robert Cowen, was of Scotch-Irish ancestry and in his day did a large stock of business at Dublin, Ireland."
So, likely names for Elizabeth's parents are Robert/Richard & Elizabeth/Harriet Cowan/Cowen/Cowin, and she was born in England between 1820-1830, possibly 'opposite Hyde Park'.
A search of the ancestry database London, England, Church of England Births and Baptisms, 1813-1906 which is from Board of Guardian Records, 1834-1906 and Church of England Parish Registers, 1754-1906. London Metropolitan Archives, London, returns no families that meet all of the given name criteria (any combination of Robert/Richard/Elizabeth/Harriet) with multiple family name spellings.
Broadening the search to all of England using "England Births and Christenings, 1538-1975," database, FamilySearch https://familysearch.org/ark:/61903/1:1:NTB1-DJM returns a few interesting results with father's name William, mother's name Caroline, and/or last name Conn, but all of those families can be traced through the 1851 census in England, and my Elizabeth was in Ohio by 1846 (see older question).
I cannot locate the family in either the 1841 England Census (which is missing several parishes 'opposite Hyde Park': https://www.findmypast.co.uk/articles/census-for-england-wales-and-scotland-missing-pieces), nor in the 1840 US census (in any state). Again, possible families in 1841 UK can be traced to 1851, with Elizabeth still with them. No potential siblings of Elizabeth have been found in the USA.
My question is, without being absolutely certain of the first names of Elizabeth's parents (and there does not seem to be more records about them), and without finding even a 'right name/right time-place' match to follow up on, how can I identify Elizabeth Cowan's birth record? Is it even possible?