At the time (pre NHS) births would either be at home attended by a midwife or doctor who would need to be payed, the other option the workhouse infirmary, would have been free the cost being covered by the local Poor Relief Fund committee.
It appears the stigma attached to workhouse births was a national thing, this taken from the London Metropolitan Archives.
Registers of Births
In many London and Middlesex poor law unions or parishes the maternity ward was in the workhouse rather than in the workhouse infirmary. After 1904 a birth certificate should not state that a birth took place in a workhouse giving instead the street and number as the place of birth, for instance, 4 Kings Road, St. Pancras, instead of St. Pancras Workhouse. Alternatively a euphemistic name might be used such as Twyford Lodge for Willesden Workhouse Infirmary or The Lodge, Bancroft Road, for Mile End Workhouse.
Registers of Baptisms
Sometimes Boards of Guardians records include registers of baptisms in the workhouse. Otherwise baptisms solemnised in the workhouse may be recorded in the parish registers of the local church
Colin