I have looked on Ancestry.Com London Baptism, Births and England records for John Phillips b 17 Sept 1797 London as per Basil data but can't see a record. As you know there is 1 Ancestry.Com private family tree which has John Phillips 1797-1859- did you contact the owner?
Also looked on London Lives 1690-1820 website for John Phillips, James Phillips, Goldsmid, Leopold- hundreds of Phillips; few Goldsmid and 1 John Leopold Jan 1790 Middlesex Sessions but insufficient data to give any further guidance.
Noted Felix Leopold married in 1846 to Julia Abrahams on JewishGen UK Data base 1851 Census but nothing earlier.

Any other thoughts regarding where to next?