Reading a bit further, it seems that the "Dog on the Tuckerbox" poem was originally called Bullocky Bill, and dates from 1857. So it predates all of Jack O'Hagen, Jack Moses and Banjo Paterson by quite a few years. All of these people drew on the work of someone who used the pen name "Bowyang Yorke".

Lazy Harry's appears to be later still, and pretty much unrelated.