Back in the 1860s, it was the responsibility of the householder to report deaths to the police and for the police to determine if it was a sudden death and/or a suspicious death in need of investigation. So whoever was the head of the household where Frances was at the time of her death would have contacted the local police, who in turn would have contacted the local Police Magistrate who would have needed to determine if the local medico had been treating the patient, and if that medico was prepared to certify a cause of death. Then the Police Magistrate likely would issue a burial order to the local funeral directors and a funeral would be carried out. Then the funeral director would follow up and get all the details he could regarding the info he needed to help complete the registration process.
So, it seems that Frances was in Mudgee with two little girls, and that she was known there as Frances FIRTH, and that the locals believed Frances was their mum. The locals did not know the name of Frances husband, nor when or where she had married, so it is fair to assume he was not around at the time. But someone has given that she was from Ireland, so we need to assume they heard her accent in her spoken words.
This is not an unusual story for the era. the 1860s saw far more people arrive at the various seaports along the coast than for all the years of the convict era (1788-1840 for NSW, and 1788-1853 for Tasmania). It was the gold fever that attracted so many to our shores in the 1850s and 1860s.
Perhaps I should suggest you have a read of the following papers prepared some years ago now, but still a valid reference source. It shows some of the issues facing females who were without their husbands. Divorce did not come to NSW until 1873.
http://www.aifs.org.au/institute/seminars/finlay.htmlCheers, JM