Hi koala, would like to help but a couple of questions and need some more info/clarification if possible.
Found Edward Henry Jones & Annie Jane (James) Jones in Ancestry 1 World Trees -- I couldn't see who made the entry. However, if it is you, then you already know he had 10 children,
1st son, Edw. Henry b. 1856 but doesn't say where born followed by 9 more children all born in Australia between 1858-1874. If you don't have the children's names, birthdates, will get it for you.
Some statistics shown in "Trees" are a little different than what you have:
Shows Edw. Henry b. 1822 (not 1827)
and marriage in 1852 in NY (not 1854)
Do you know when Edw. Henry left Wales/England for NY?
Was he a child or adult? Anyone travelling with him?
Did Annie come at the same time i.e. travel with him or separately or join/meet him in NY?
In 1850-1854 there are about 1000 each of Jones & James per year arriving from England so it's a lot to go through without specifics. Passenger records often just say Mr. Jones or Miss James or E. Jones, etc.
What was Edwards profession? If he had a profession like Dr., Lawyer, etc. he might appear in a directory or other reference.
In any case, they were not in US for 1860 Census, so may be hard to find anything about them unless 1st son Edw. Henry was born in U.S.
I've never searched for emmigration data from US to Australia --don't know how records kept, but I'm sure someone on the emmigration to Australia board might be able to help you with that.
Kathy