Births were mandated to be registered after Sep 1837, but people didn't always comply right away.
Parish registers will show baptisms; I've seen a few that include both birth and baptism dates (invaluable!!).
Before Sep 1837, parish registers were the only "registrations" of any kind for birth/baptism. Children were usually baptised a month after birth, but not always. I happen to know my grandmother was born in October and not baptized until the following January.
If you haven't already done so, PLEASE visit your local LDS Family History Center (FHC). They are invaluable resources. The Mormons filmed original registers and other parish documents and these are available on microfiche and/or microfilm that you can order sent to your FHC and review there.