Is it obligatory to inform one's brother directly about a correction to previous information regarding the uncovering of awrah, or is it permissible to inform him indirectly?

The affairs of Muslims should be interpreted favorably, and it is not permissible to accuse them of wrongdoing without evidence, especially one's mother. You did well by seeking forgiveness from your mother, and you must fulfill the conditions of repentance. You are not required to inform your brother of what you accused your mother of; rather, be sure to praise and commend your mother in front of your brother, and make abundant supplications for her.