You have posted on the Ireland board, not the ideal place.
Suggest you research the details further within Australia. You need to work it back from your father using Australian resources and certificates to confirm where they came from in England, Scotland or wherever as though I see a tree on Ancestry with similar details they look spurious, with the Victoria, Australia, Passenger List suggesting William Thompson junior was Scottish, a London school admission and a Barnsley Baptism. Ancestry hints lots of records with the same names but many are invalid and are hinted as others have the invalid records in their tree.
It may be the Australian stuff is ok back to William & Catherine Luddy but you need something confirming their parents names eg a death or marriage Cert and something confirming where he came from rather than stabbing in the dark pre-emigration.
Libraries may have access to Ancestry if you don't have a sub.
For what it is worth 1837 marriage cert image
https://www.ancestry.com.au/discoveryui-content/view/90849535:62237https://www.ancestry.com.au/family-tree/person/tree/171680903/person/212228186132/facts is one that suggests William and Mary emigrated to Australia and had the son who married Catherine Luddy - in Victoria.
However, the only William Thompson with a mother Dale was born Guisborough 1850.
The one born 1848 Barnsley had mother Mahon or Ashirst.
For England the GRO's own website contains mother's maiden surnames from 1837 which are not in the original period indexes.
From that 1837 marriage there were 2 daughters before 1841 Frances and Hannah appearing on the census
https://www.ancestry.com.au/discoveryui-content/view/11924690:8978 By 1851 Frances was a 13 year old servant in Brompton by Sawdon, Scarborough and not sure where parents were, there were a lot of Thompsons in the Helmsley area in the census.