Hi Colin,
I found the Marriage in NZ to Mary and am very happy you have sent for the Cert.
There is nothing on FreeBMD for Edwards birth reg. abt 1838 BUT a possibility that would not have possibly made civil reg

From FreeREG;
Edward TOMPKINS
Baptised, 15 June 1837, St Peters, Wolverhampton, Staffordshire
Father, JAMES, occp. Servant
Mother, LOUISA
Trish
EDIT;
James may have married Louisa WOODEN in Warwickshire just after the Birth of Edward.
There is more I have found but until you have the Marriage Cert it is just all supposition.
They have several children born in Warwickshire, BIRMINGHAM
LOUISA, born 1842, christened 1844
JAMES, 1844
GEORGE, 1846