He wants to go back to his country but there is war there; is his wife obliged to obey him?

Firstly: 

Staying in a kafir country is not permissible unless certain conditions are met, the most important of which is that the person be religiously committed and thus protected against whims and desires; he should also have sufficient knowledge and insight that will protect him against specious arguments; he should be able to practice his religion openly; and he and his family should be safe. For more details please see the answer to question no. 13363 and 27211. 

Secondly: 

The wife is obliged to obey her husband and move with him wherever he moves, so long as she did not stipulate that her husband should not take her out of her homeland or make her travel to a specific country, or so long as moving with him will not cause her any obvious and considerable harm, such as if he wants to move to a country where there is war or it is thought most likely that she will encounter harm such as imprisonment and the like. 

Ibn Qudamah (may Allah have mercy on him) said: He may travel with her, because the Prophet (blessings and peace of Allah be upon him) used to travel with his wives, unless it is a kind of travel in which there is fear (of harm), in which case she does not have to do that. 

End quote from al-Mughni (7/223) 

And it says in Kashshaf al-Qina‘ (5/187): The husband may travel with his wife, because the Prophet (blessings and peace of Allah be upon him) and his Companions used to travel with their wives, unless the travel involves fear or there is danger on the road or in the land to which he wants to travel. In that case he does not have the right to take her with him without her permission, because of the hadeeth, “There should be neither harm nor reciprocating harm.” Or if she stipulated that she should stay in her own land, in which case she is entitled to that condition, because the Prophet (blessings and peace of Allah be upon him) said: “The most deserving of conditions to be fulfilled are those by means of which intimacy became permissible to you.” End quote. 

She does not have the right to refuse to accompany her husband because of the children’s education, because education may be acquired in various places and through various means such as distance learning and so on. So it is not permissible for her to disobey her husband for the sake of her children. 

Thirdly: 

You have to obey your father and travel with him if he wants that, so long as your travelling to his country will not pose a danger to you. 

It should be obvious to you that the father has rights and a high status, and Allah has commanded us to honour our parents; He has mentioned that alongside obedience to Him and affirmation of His Oneness (Tawheed). Claiming that the father is not thinking of his children’s best interests is mistaken thinking in most cases, because of the natural compassion that the father has for his children, but what the father thinks is in the children’s best religious or worldly interests may make him overlook this compassion. Allah has commanded the father to protect his children and guard them against that which may lead to doom and loss, as He says (interpretation of the meaning):

“O you who believe! Ward off yourselves and your families against a Fire (Hell) whose fuel is men and stones, over which are (appointed) angels stern (and) severe, who disobey not, (from executing) the Commands they receive from Allah, but do that which they are commanded”

[al-Tahreem 66:6] 

And it was narrated from Ibn ‘Umar (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said:

“Each of you is a shepherd and each of you is responsible for his flock. The ruler of the people is a shepherd and is responsible for his flock. A man is the shepherd of his household and is responsible for his flock. A woman is the shepherd of her husband’s house and children and is responsible for her flock. The slave is the shepherd of his master’s wealth and is responsible for it. Each of you is a shepherd and each of you is responsible for his flock.”

Narrated by al-Bukhari (7138) and Muslim (1829). 

And al-Bukhari (7151) and Muslim (142) narrated that Ma‘qil ibn Yasar al-Muzani (may Allah be pleased with him) said:

I heard the Messenger of Allah (blessings and peace of Allah be upon him) say: “There is no person whom Allah puts in charge of others, and when he dies he is insincere to his subjects, but Allah will forbid Paradise to him.” 

So it is no surprise that a father would flee with his children from the land of kufr, fearing corruption and deviance for them, and that he would prefer soundness of religious commitment to worldly interests. If his move to his homeland is safe, then all the members of his family are obliged to obey him and accompany him. 

And Allah knows best.