Is it permissible to marry a girl with only her mother's consent, given that she has been estranged for years from her irreligious father, and is the father's consent obligatory in this case?

The guardian is a condition for the validity of the marriage, so it is not permissible to marry a girl except with the permission of her guardian. Her mother's consent is not sufficient because guardianship belongs to the agnates (male relatives), and the most rightful person to give her in marriage is her father. If the father refuses to marry her to a suitable suitor, the girl can refer her case to the Sharia court or to Islamic centers in the West so that they may marry her off if her guardian's obstruction is proven, based on the saying of the Prophet, peace and blessings be upon him: "The ruler is the guardian of one who has no guardian." If marrying her is not possible, then seek someone else.