Kerryn,
Wow! You are trying to get somewhere with lots of "iffy" data! I would say your best hope is to eventually find a document in Australia which happens to mention from whence he came in England. One thing I've learned over here is to not believe it when you are told something like, "Naturalization forms prior to the year XXXX did not include counties of origin," for example. I have run across documents where a clerk happened to insert more than the required detail. This could occur on a document from much later in his life (or death), too, such as a cemetery record.
Other than that, I just do not see how you are going to be able to locate a birthplace, which is necessary in lieu of a more exact year of birth. I think you said something about one of them being apprenticed somewhere. Some of those records can be accessed. Look for the particular England-wide guild to which he might have belonged once he became a journeyman, if that occurred prior to leaving for Australia.
On Find My Past I found the following possibilities (added to those on Familysearch.org previously mentioned), but those searched were based on the assumption that 1780 was a pretty firm date.
1779 in Kirkheaton, West Riding, Yks to Joseph and Hannah Dawson. (Kirkheaton is near Huddersfield.)
1779 in Huddersfield to William Dawson
1780 in Weardale St. John's Chapel in County Durham to Joseph Dawson. (It looked to me, however, that this Joseph was still there at age 60 on the 1841 census.)
I sure wish you good hunting! If you find something you would like me to check out further, let me know.
Linda