Here is the 1900 census information which I believe you may already have. Is this the correct family?
1900 US Federal Census Philadelphia Ward 5, Philadelphia, Pennsylvania
230 Pine Street
-Henry Cohen 54, b. Apr 1846; married 25 yrs.; b. England, Parents b. Russia; Immigration yr: 1870; Naturalized citizen; Occupation: Shirt Maker; rents home -Rachel Cohen 44, b. Dec 1855; married 25 yrs.; Mother of 6 children/5 children survive; b. NY, Parents b. Russia -Sadie Cohen 18, b. Feb 1882 NY -Moses Cohen 16, b. May 1884 NY -Abram Cohen 10, b. May 1890 Canada-Eng. -Mildred Cohen 8, b. Mar 1892 Canada-Eng.
Possibly, his immigration to the U.S.:
Ship Name: City of Antwerp Port of Departure: Liverpool, England Port of Arrival:New York
Name: Henry Cohen Arrival Date: 9 Mar 1872 Birth Year:abt 1846 Age: 26, male, Occupation: Clerk Ethnicity/Race-/Nationality: English Place of Origin: England Destination: USA
NARA Series/Roll: M237_353; Line: 47; List Number: 166
This may or may not be your ancestor, Henry Cohen in the English census:
1851 England Census
Name: Henry Cohen Age: 5 Estimated Birth Year: abt 1846 Relation: Son Father's Name: Samuel Cohen Mother's name: Mary Cohen Gender: M (Male) Where born: Whitechapel Civil parish: St Botolph Aldgate County/Island: Middlesex Country: England
Registration district: East London Sub-registration district: St Botolph ED, institution, or vessel: 1b Household schedule number:52 Piece: 1524 Folio: 42 Page Number: 17
Household: -Samuel Cohen 32, b. London Aldgate; Occupation: Clothes Dealer -Mary Cohen 26, b. London Aldgate -Lennis (or Lewis) sp? Cohen age 7, b. Whitechapel; Scholar -Henry Cohen 5 (c. 1846), b. Whitechapel; Scholar -Phillip Cohen 3, b. Aldgate -Nathan Cohen 6 Mos. old, b. Aldgate -Ellen Brian 20, b. Ireland - Cork; Occupation: House servant
The National Archives of the UK (TNA): Public Record Office (PRO), 1851; Kew, Surrey, England
However, the U.S. census indicates Henry's parents were born in Russia and the census from England does not reflect that.
You had written Henry Cohen was from Liverpool (Lancashire), but was he born there?
I'd suggest obtaining a copy of his obituary or his death certificate to find out the names of his parents. Another possibility is to contact the cemetery for his burial records. Once you have his parents' names then it will be easier to discover information about his early life.