All perfect praise be to Allah, The Lord of the Worlds. I testify that none is worthy of worship except Allah and that Muhammad is His slave and messenger.
First of all, we would like to caution you that ruling a Muslim to be an apostate is a very serious matter that must not be taken lightly. In principle, a person whose Islam has been proven with certainty cannot be judged otherwise except with certainty. We cannot declare a specific Muslim to be a disbeliever unless certain conditions are fulfilled and certain impediments are absent. So beware of taking a risk in this regard.
Assuming that your mother had apostatized –Allah forbid– and then repented and reverted to Islam before her ‘Iddah expired, then her marriage to your father would still be intact. However, if she did not repent and revert to Islam before her ‘Iddah expired, the marriage would be annulled and it would be impermissible for your father to live with her.
In this case, the separation of spouses falls under the jurisdiction of the Sharee'ah Court. As for you, you are not required to live with them so that they would not be seclusion together, and it is not obligatory on you to accompany your mother when she travels. However, you should advise her kindly in this regard and clarify the matter to her. It goes without saying that you should exert your utmost effort in calling your mother to the truth and strive for her salvation and bringing her back to Islam. Indeed, this is one of the greatest acts of dutifulness and kindness you can do for her.
Allah Knows best.