Re NSW BDM online Birth records

The early ones, ie those referenced with the letter "V" are only baptismal records. Although civil registration commenced in 1856, even the NSWBDM online site recognises that it was not until 1879 that they received regular quarterly returns for BDM events from Clergy. So for births before 1879, even those from 1850's, there's reasons to suspect that there's many gaps in the NSW records. The C of E handed their bdm registry to NSW Reg Gen in 1879, that register covered 1825-1855...
From my own tree, my NSW born 19th Century forebears include bdm certificates for most, but despite much searching for decades by various family members (SAG members etc), birth/baptismal records have not been found for several, including two lasses born in Sydney in the 1850's, raised as Methodists. The NSW marriage certs, death certs and the many births of and early deaths of their children all show these two were born in "Sydney N. S. Wales", and their older and younger siblings birth/baptismal records are all found indexed online at NSW BDM. Also, the YEAR noted as part of NSW BDM reference is the YEAR that NSW BDM actually entered the info on THEIR records, ie not necessarily the year the event actually occurred. Hence from my own tree, a birth in the Wentworth area (Lower Darling River) that occurred in 1884, was not entered by the Sydney based Reg Gen's office until 1907, so it is indexed with 1907 as its indicated year, and did not come online until 2008
The NSW State Library's film collections in the family history section are extensive and I find them to be more reliable than the NSW BDM online index, especially for the "Early Church Records".
So it is possible that if HDHE was born in NSW, that the registration of his birth or his baptism was simply not forwarded to the NSW BDM Reg General, or that the Reg General did not process that quarter's return, from the civil servants in the 1850's or if baptised C of E from the registers sent to them in 1879. The 1850's rural registrars were usually part-timers, ie usually a ranked Police officer or Magistrate or Sherrif. And many of them quit and went off with gold fever running through their minds. Some quarterly returns would have been lost in transit as they were sent from rural areas via the Royal Mail carried by Cobb and Co etc and ranksacked by Bushrangers who held up the escorted coaches on their way to Sydney.
The 1850's also saw the separation of Victoria and Queensland from the Colony of NSW, so perhaps HDHE's birth/baptism registration (if in N S Wales

) may be found only in a parish record anywhere on the east coast !
I'm not familar with Welsh Records, but Jenn, have you recently tried posting a look up request on the RChat board for Wales

Cheers, JM
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