What should I do about my married brother with children who is committing adultery, knowing the danger of his affair being exposed to our elderly mother?

What your brother and cousin are doing, which is reprehensible, is forbidden (haram), especially since your brother is married, making his punishment more severe and his sin greater. Your duty towards them is to offer advice and clarify the truth in an appropriate manner. If you cannot do so directly, seek assistance through indirect means such as friends and relatives whom they trust. If you are unable to change the reprehensible act with your hand or your tongue, then change it with your heart, for the Prophet, peace and blessings be upon him, said: "Whoever among you sees an evil, let him change it with his hand; if he cannot, then with his tongue; if he cannot, then with his heart—and that is the weakest of faith."