Update, someone has just re-pointed out that 1 Union Road is a private house. Yet the mother is living there and the baby was born there. The mystery continues.
1 Union Road = Erdington House, Union Road, Erdington (Aston Union Infirmary/Workhouse)
In the 1921 census, several of the staff who work there refer to it as "Erdington House".
Erdington workhouse records are in the Archives Dept of Birmingham Central Library, but any records within 100 years are closed to researchers. There should be Religious Creed registers, which include information such as: relatives, last address, and date of discharge or death.