Is the wife obligated to obey her father and not contact her non-Muslim mother, or is she obligated to contact her mother, especially since she is her only daughter, knowing that her father will not forgive her in this world or the hereafter if she contacts her?

Islam has granted great rights to parents, even if they are disbelievers. However, this applies specifically to the an original disbeliever. The case of an apostate mother—meaning one who was Muslim and then left Islam—is different. Apostasy is more severe than original disbelief. The jurists have agreed that the ruler executes the apostate and does not allow them to persist in their apostasy; they only escape execution by returning to Islam. This ruling is to be carried out only by the ruler or his deputy. Accordingly, the wife must boycott her apostate mother, unless doing so is a means to invite her back to Islam.