Back to Liverpool baptisms on Lancs OPC:
as well as James Fishwick baptised in 1781 there are
9 Nov 1783 St Nicholas, Liverpool
Janet Fishwick - Daughter of John Fishwick & Catharine
Born: 7 Oct 1783, Abode: Old Hall street, Occupation: Taylor
16 Jan 1786 St Nicholas, Liverpool
Elizabeth Fishwick - Daughter of John Fishwick & Catharine
Born: 12 Dec 1785, Abode: Old Hall street, Occupation: Tailor
10 Jul 1795 St Nicholas, Liverpool
Catharine Fishwick - Daughter of John Fishwick & Catharine
Born: 16 Jul 1794, Abode: Spellow Alley, Occupation: Tailor
Maybe Jane Fishwick of Gregson St is this Janet Fishwick, sister of James Fishwick and the Elizabeth that married Thomas Holland.
I would bet that your John Fishwick is grandson of John Fishwick, tailor of Liverpool, but could do with his baptism to settle it.

Barbara