Woman trained to help other women in child birth' is called -

Midwife: a person, typically a woman, who is trained to assist women in childbirth.
Matron: a woman in charge of domestic and medical arrangements at a boarding school or other institution.
Nurse: a person trained to care for the sick or infirm, especially in a hospital.
Wet Nurse: a woman employed to suckle another woman's child.

Source: Oxford Dictionary