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You are fortunate the marriage took place in Nova Scotia because the marriage registration gives lots of details that might help.
John William (not William John) Heaffey married 19 Feb.1876 (Presbyterian) to Catherine Harper (age 21). His details say that he's age 28 and a baker, born London, son of John William (fireman gas factory) and Catherine.
In 1891 census (Battersea, England) John W. Heaffey, age 42, born Westminster, London.
In 1901 census William John Heaffey, age 52, born Malaw Ireland and wife Katherine born Hona, Scolia, Halefax [Nova Scotia, Halifax] according to Ancestry.
Assuming for the moment that almost all records say John William/William John was born in London- have you tried searching for him and especially his parents in English records? The surname might be under something like Heaphy, etc. You have the names of both parents and father's occupation from marriage details so you might be able to find the family. Even if J.W./W.J. was born in Ireland the family could have moved to England when he was young.