A Guide to Using the Records of Civil Registration from the Society of Genealogists says:
The records of civil registration in England & Wales, which commenced on 1 July 1837, relate to the birth, marriage and death of an individual. ...
In England & Wales, up to that time, the government had relied very much on the church to register its population but it realised that it was not a complete record i.e. not a full listing of the population. Therefore, a single tier registration system was introduced, based on the administrative poor law unions, which had been set up in 1834, and previously the administrative hundreds. These became the registration districts. Births, not baptisms, & deaths, not burials, were recorded as well as marriages. Parish and nonconformist baptism & burial registers were still completed at the same time that the new civil registration system began.
A birth that led to a baptism in Bulford, Wiltshire, England in 1823 is almost 15 years too early to have been registered. Some baptism records record the birth date alongside the baptism date. So I think your next step should be to try and locate a transcript and image of the 1823 baptism record.
If I find a baptism record on the 6th April 1823, and it does not include a birth date, then I record the birth date as being "Before 6 Apr 1823".