Lodges kept their own records, different lodges had different ways of keeping records, and it depended also on who was the then current person responsible for the record keeping.
Formal Adoptions did not commence in NSW until December 1923, so there was no formal authority for the NSW Supreme Court to become involved in the (fostering) process
Any person could and can still become known by a name that has no bearing on the name on their birth certificate. It is not compulsory for a formal deed poll to be arranged. It certainly was not even a usual process back before WWII.

When a child started school, it may have been a decision by a class teacher to use the surname of the new parents for that child, particularly if other siblings were also at that same school.

No BDM paperwork would have been needed. Just because there's no NSW marriage or death registration found at the online index ... well it would be flukey but there are known flaws in the online index. Could the person have married elsewhere ....

or become known by her partner's surname?
The Child Welfare Act has been amended many times .... here it is in 1924
http://www.austlii.edu.au/cgi-bin/viewdb//au/legis/nsw/num_act/cwa1924n69222/My rellie will think more about this and contact me tomorrow.
JM