I have researched all the McBurney lines in NSW as fara as the publicly available info will allow. I don't have any of the story of your family but I can email you the tree of your line I have constructed. It shows that Edward may have had a brother Robert and a sister Mary Jane as deaths for these 2 were registered with parents shown as Robert & Mary.
My line comes from Co. Armagh, Northern Ireland (and certainly Scotland prior to that) but I am not able to establish a connection with yours.
Email me at smithy57 att bigpond dott com with your direct lines, I'll fit you in and email you the tree back. You can take a gedcom file, right?