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.
There is no doubt that a mother by nature is merciful to her children, especially to her daughters. So, what you mentioned about your mother – if this is the truth – is strange and it is an evil matter that is not permissible. In principle, a Muslim is innocent until proven guilty, so it is forbidden to suspect a Muslim without evidence. For more benefit, please refer to Fatwa 81748.
Accusing a Muslim of Zina (fornication or adultery) is especially more sinful and abominable and it is a cause of being cursed. Allaah says (what means): {Indeed, those who [falsely] accuse chaste, unaware and believing women are cursed in this world and the Hereafter; and they will have a great punishment.} [Quran 24:23]
The accuser is required to produce evidence for his claim, namely four witnesses. Otherwise, he deserves the Hadd (prescribed punishment in Islamic Law) of 80 lashes. Allaah says (what means): {And those who accuse chaste women and then do not produce four witnesses — lash them with eighty lashes and do not accept from them testimony ever after. And those are the defiantly disobedient.} [Quran 24:4]
It is also a clear evil that your mother goes to fortunetellers. This itself is a great sin. If she believes what they say, the matter is even more serious. It is established that one of the wives of the Prophet narrated that he said: “Whoever goes to a diviner and asks him about something, his prayers will not be accepted for forty nights.” [Muslim] Moreover, Abu Hurayrah narrated that the Prophet said: "Whoever comes upon a woman during menses, or comes upon his wife from the posterior, or goes to a soothsayer and believes what he says, he has indeed disbelieved in what was revealed to Muhammad.” [Ibn Maajah] For more benefit, please refer to Fatwa 129376.
Another great sin is what you mentioned about her not performing the prayer, as this is a very serious matter. Some scholars are of the view that the person who forsakes the prayer, even from laziness, has left the fold of Islam. Therefore, you are obliged to give advice to your mother in a wise and gentle manner and warn her of the severe punishment of Allaah. You may seek the help of some righteous people in advising her with hopes that she will repent. This way, you will be doing her a great favor.
If she persists in what she is doing, then this does not prevent you from being kind and dutiful to her as this is her right over you even if she mistreats you and is wrong. If you mistreat her in the least, you will be undutiful to her at any rate. For more benefit, please refer to Fatwa 90264.
As for alienating her, some scholars permitted alienating a sinful parent if there is probable cause to believe that there will be a benefit in doing so as clarified in Fatwa 89611.
Shaykh Ibn ‘Uthaymeen was asked: “Is it permissible to desert Muslim parents if this is for a Sharee'ah-based benefit? He answered: “Yes, if there is a Sharee'ah-based benefit for Muslims parents in deserting them, then it is permissible to desert them, but this does not necessitate breaking kinship ties with them. You are still required to keep ties with them, like by providing them their food, drink, accommodation, and the like.” [End of quote]
In most cases, there is no benefit in deserting one's parents, so it is better to avoid it. Nonetheless, if it is your mother who starts deserting you, then she is the one who is sinful.
Allaah Knows best.