Many thanks, Trish!
I had found the service record of Ernest Rope when I first looked at this last year on a general hunt for any Cornhills in Australia. It didn't occur to me to look for his death to find an indication of his parentage - you rarely find such info on a UK death cert for an adult. But as you say, there it is - father James J.
Not conclusive, of course, as the names have been reversed and the surname isn't in the index entry. But Joseph James was the son of another Joseph, so may have used his second name - just like Ada Mary, who is Mary A at birth but Ada M everywhere else.
So that narrows it down - Joseph James is likely to have been still in UK in 1883 when John Thomas Cornhill died, and Wallace too, when his wife and son died in 1883/4 - and both were in the 1881 UK census with their families. But they were in Australia by 1887 when Ernest was born.
Of Ada's other two children Lilla May married Alfred Turner, and her death index listing gives her parents as Arthur and Agnes May. As the death particulars were probably supplied by her husband, the Agnes May could be simple error or a bad memory, but Arthur is unlikely to be confused with Joseph or James. Susan died in infancy, with just her mother listed, so no further clue there.
So what I need to prove the link is the death cert or a transcript for Ernest, and perhaps his marriage cert as well for additional confirmation, the marriage cert for Joseph James and Ada and the marriage cert for Lilla and Alfred to see if her father's name really was Arthur.
Which still doesn't answer the question of how and when Joseph James - and almost certainly Wallace - came from the UK to Australia. Wallace is the clincher in many ways - only one Wallace Cornhill (plus his son who died in infancy) appears in the UK GRO indices and there is only one listed in the entire range of census returns. And only one Wallace on the IGI - the same one.
Never mind - if the certificates show that this IS my Joseph James, I may have to be content with simply noting that "Emigrated to Australia between 1883 and 1887"....

Thanks to everyone for the help!