What should a girl do whose father gave her the choice between the niqab and university, after she wore the niqab without his permission, with his complete refusal and threat to cut off his relationship with her?

Obedience to parents is limited to what is good and permissible. It is not permissible to obey anyone in an act of disobedience to the Creator. Covering the face is obligatory according to the most preponderant opinion, so it is not permissible to obey parents in removing the niqab, except out of necessity or a need that is considered equivalent to a necessity, such as a real harm or damage resulting from wearing it, or being prevented from essential university studies. However, if there is no necessity, then covering the face must be maintained. You should try to convince your father gently and politely about the necessity of the niqab, and seek your mother's help in this regard.