WOW! Thank you so much for your help, already so many good questions.
First - "muddled online family tree" that's mostly my doing. The version on ancestry is cribbed (by them) from the one I've been doing on family search, which almost always contains a link or a reference to primary documentation.
Second - I am aware of the town of Hanmer - I've even visited it long ago. I've reviewed the various Hanmer family trees both there and in Myddle from the relevant time period, but there's no obvious contenders, like for instance a "Leah Hanmer" born about 1790 - but I could look again.
Lydia Jones' children are listed in her death certificate, which can be viewed here:
https://drive.google.com/open?id=0B2qT_gSEY6W6UWpzWi1VWFdLZkE (and here)
https://drive.google.com/open?id=0B2qT_gSEY6W6VW9nRDFYam92Sm80S3FrWjF2QVM1TUNqOThBThere is a John and Ellen Williams family in Oswestry who had two children (Owen and Mary) in about 1815 (eg
https://familysearch.org/ark:/61903/1:1:J33D-R97). There are also numerous Williamses in Wolverhampton who have roughly the right names and dates, but nothing compelling.
John Jones (senior) and John Williams are both listed as labourers in the 1845 marriage certificate. John Jones was given as a miller in his marriage certificate, but in Australia worked as a sweep, a miner, and possibly a sailor. His son worked as a thresher for a time before becoming a farmer.
Earliest written record of Handmer-like name is in 1856 (
https://drive.google.com/open?id=0B2qT_gSEY6W6bkVGZEJVaThIcDg). John Jones' name is given here as John Amer Jones, and his daughter (his last child before Lydia's death) as Mary Ellen Hangmer Jones. Confusingly, Mary Ellen was renamed Lydia after her mother died and lived with Leah for a few years before accidentally drowning in about 1863. The informant in Mary Ellen/Lydia's birth record is Leah, who was about 16 at the time and barely literate - so spellings vary throughout her family's life. By 1875 (when the next generation survivors resurfaced as adults) they were no longer Jones.
I think that answers most of the questions thus far, I'll check back though!