Praise be to Allah.
All the individuals involved in this situation are wrong and are committing sin, and perhaps you have been the most wrong of all, for several reasons. You are the one who has been writing these sinful love letters for your mother; you have been concealing this from your father and have even lied to him, telling him that nothing is going on; you did not advise your uncle to refrain from doing haraam things, namely going against the teachings of his religion and betraying his brother. You did not confront your mother frankly and make her give up this foolish and haraam action – in fact you regarded her action and her justification for it as just an opinion, as opposed to your opinion that her action is not justified, when in fact the issue is one of whims and desires as opposed to a shar‘i ruling. You even tried to spare her feelings so that she would not be upset if you refused to do her bidding and write haraam letters to your uncle and correspond with him in her name.
All of that leads us to fear that you are the most sinful of all the individuals involved. You could have put a stop to this evil immediately, by confronting your mother and telling her frankly that what she is doing is haraam, that it is not permissible for her to carry on with it, that her justifications for it are not acceptable in terms of either sharee‘ah or reasoning, and that you would inform your father if she continued this sinful relationship with your uncle. You could also have put a stop to this evil by confronting your uncle with his evil deeds and telling him that a person like him could not be entrusted with people’s honour; by threatening him that if he continued this action, you would tell your father about what he is doing and you would prevent him from entering your house. You could also have put an end to this evil by telling your father about the details of the matter so that he could do what Allah has enjoined upon him of advising his family members. We do not advise you to tell your father unless your mother or your uncle persist in this sinful relationship between them. If they give up that relationship then there is no need to tell your father about it. But if they persist in it – or if one of them persists – then you do not have the option of remaining silent; rather you must – whilst also continuing to advise and exhort them – tell your father so that he can put a stop to this sinful relationship, even if that leads to him divorcing his wife or cutting off ties with his brother and banning him from entering his house.
We are doubtful about your saying that your father knows about your mother’s relationship with your uncle and that the matter is out of his hands. If we assume that the matter is as you say, and that your father knows what is really going on between his wife and his brother, but he is keeping quiet about it and is not doing what the situation requires, then this is a case of cuckoldry. He has the power to advise his wife, or to shun her, or to hit her; he also has the power to ban his brother from entering his house, or to cut off communications in his house. What makes us doubt that he knows about this relationship is the fact that your mother has been trying to conceal it from him, and he asks you and you deny it. As for his stress, it may be an indication of his doubt; it does not indicate that he knows what is really going on, especially since you say that your mother does not talk to him face-to-face when he visits you. We are certain that if he knew about this haraam relationship, he would act as is befitting for the head of a religiously committed family, as you claim to be.
One of the clear rulings of sharee‘ah is that it is haraam to cooperate and help in sin and transgression, or to obey if doing so involves sin. There is no obedience to anyone if it involves disobedience towards Allah, even if the one who is issuing these orders is a mother or father; rather obedience is only in that which is right and proper.
Allah, may He be exalted, says (interpretation of the meaning):
“Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression. And fear Allaah. Verily, Allaah is Severe in punishment”
And it was narrated from ‘Ali that the Messenger of Allah (blessings and peace of Allah be upon him) said: “There is no obedience if it involves disobedience towards Allah; rather obedience is only in that which is right and proper.”
Narrated by al-Bukhaari, 6830; Muslim, 1840.
Based on that, it is not permissible for you to hesitate about refusing to obey your mother and doing that evil action with regard to the sinful relationship between her and your uncle. Her justification for that is not acceptable and it is not even worth paying attention to.
And Allah knows best.