What is the ruling on migrating to another country to preserve one's religion, given the mother's strong rejection of this, and is it obligatory to fulfill a marriage promise to a girl whose life may collapse if the marriage does not take place?

If a Muslim is unable to practice the rituals of his religion or fears for himself and his family from fitnah (temptation/tribulation) in lands of sin and disbelief, then emigration (hijrah) becomes obligatory upon him, even if his mother disapproves. However, if he is able to practice his religion and feels safe, then emigration is recommended, not obligatory, and obedience to the mother in this case takes precedence over emigration. There is no harm in marrying the girl he promised, especially if she is pleasing in her religion and character.