What is the ruling on an oath when a mother makes her daughter swear by saying "Wallahi" or "Wallahi Al-Atheem" to find out something concerning her daughter's husband, while the husband has asked his wife not to tell anyone?

Your mother has no right to ask you about matters concerning your husband that do not pertain to her. You can apologize to her by saying that your husband has forbidden you from disclosing anything personal about him. If you fear a greater harm, then use allusive speech (ma'arid). If you have done this previously, we hope that there is no sin upon you, and no expiation (kaffarah), because this oath is not a binding one that requires expiation.