All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and messenger.
First of all, we appreciate your keenness on staying firm and adhering to the rulings of the religion; may Allaah reward you and enable you to perform good deeds and acts of obedience, and may He guide you even more.
We ask Allaah to guide your mother to the right path, and we recommend you to supplicate Allaah for her. This is one of the greatest acts of kindness and obedience to her. Perhaps Allaah will guide her thanks to a good supplication from you, and Allaah is able to do all things.
Also, you should advise her in a wise and gentle manner, especially as you mentioned that she claims to believe in Allaah but, in fact, she disobeys Him. Indeed, faith necessitates performing acts of worship, as Allaah says (what means): {Except for those who have believed and done righteous deeds and advised each other to truth and advised each other to patience.}[Quran 103:3]
You should also seek the help of righteous relatives and pious people whom you expect that she will listen to and who could influence her. For more information, please refer to Fataawa 87672 and 108184.
There is no doubt that it is not permissible for a Muslim woman to marry a non-Muslim man, and if this happens, the marriage is invalid. For more benefit, please refer to Fatwa 87151.
Your mother staying in this illegitimate relationship with that man is nothing but fornication and involvement in receiving boyfriends for unlawful sexual enjoyment. Therefore, he is not a Mahram to you and you must consider him as a non-relative man. You should neither allow him to be in seclusion with you nor take off your Hijab in his presence.
Therefore, you should discuss the matter with your husband to find a solution so that you will not stay with your mother while waiting to move to live with him.
On the other hand, if your mother is an apostate, then marrying an apostate is invalid, whether the husband is a non-Muslim or else, as stated by the jurists. Zakariyya Al-Ansaari from the Shaafi’i School, said in Asna Al-Mataalib: “It is not permissible for anyone to marry an apostate. To explain, a Muslim is not allowed to marry an apostate woman because she is a disbeliever whose disbelief is not allowed under Sharee'ah just like an idol worshipper. Also, it is not valid that an apostate woman be married to a non-Muslim because she has an origin of Islam inside her. Likewise, she is not allowed to be married to an apostate because the purpose of marriage is continuity while it is not continuous in this case (because an apostate is to be killed as per the prescribed punishment for apostasy)…”
Allaah Knows best.