Attitudes to adoption/illegitimacy have changed over the last 70 years - you would need to check other Poor home births to see if it was common practice at that time for the administration to register the birth etc - if your ancestor was the only such case then look for reasons (incapability, forced (or voluntrary) adoption of child etc) - my own guess (empasis on guess) is that I would expect the Poorhome to do all the paperwork involved in registration and that many such births would be registered by them. Giving birth at the poorhome was probably the last resort (e.g. no family support) - the mother would not be able to keep the baby and her employment, thus giving up the baby at birth might seem the only choice she had.