Thanks everyone for such a quick response. In reply to a few questions ....
1. The Francis & Mary connection.
I have obtained marriage and death information for Francis & Mary and their 9 children, and have followed most of these children through their lives via electoral rolls, etc. The only one who could possibly have changed his name to Ernest was Edward John, born 1875, and whose last record I could find was before the 1913 wedding, in 1909, when he is listed on the electoral rolls as a farmer. It isn't impossible that Edward John changed his name to Ernest McQuillan, gave up farming for cooking and dropped his age 5 years, then ran away with my grandmother to NSW, but I can't see any way to test that.
I live in Sydney (though a long way from Kingswood!) so I paid a couple of Victorian researchers to follow up more, by checking school records in Stawell & Carlton where they lived, police records and Francis' will (as you suggest, Ros). We couldn't find any trace of an Ernest - though, curiously, Francis' will wasn't granted probate until 1934, 32 years later.
The only possible (very vague) connections we found is that Francis Hargrave's father was a weaver, and one of his sons was a cook, same as Ernest. But that doesn't lead anywhere.
I also have been in contact with a descendant of one of Mary McQuillan's siblings, and he has quite a bit of family information, but nothing that identifies a possible Ernest.
2. The second marriage
Ernest married again and had one child. She was 7 when his second wife died and 12 when he died, so someone had to be looking after her for a while. She eventually went to England, married and moved to Rhodesia in 1956, and I haven't been able to trace her further. But her husband's name was Asgill-Tucker, quite an uncommon name, and there are a few Asgill-Tuckers in Australia now. Quite possibly the family migrated from Rhodesia after independence, but so far I haven't been able to contact any possible descendants.
3. The Jervis Bay connection
He was apparently a cook at the Jervis Bay Naval station, but was not a service person (according to an email from the Navy) and he left about the time it started up - so I assume he worked for the construction company. I haven't been able to find any record of all of this apart from my mum's memories and the notation on his marriage certificate.
Thanks everyone for helping out with suggestions. Thanks Ros for your wonderful help to a newcomer. Jude, I think you are right about the McWilliam/McQuillan thing being a mis-hearing, though when searching for McQuillan I usually also try McWilliam just in case.