Is it permissible to travel for 'Umrah with one's wife without informing the sick and elderly mother, in order to avoid her anger or feeling of sorrow, given that there are siblings who care for her?

May Allah reward you for your efforts in reconciling your husband and your mother, and for your appreciation of your mother. As for your trip with your husband for Umrah without informing your mother, the matter depends on what serves the best interest. If you see that the best interest lies in not informing her, so that she does not grieve, and you are certain that she will not learn of your trip from anyone, then there is no objection to traveling without her knowledge. However, if you know that the best interest lies in informing her, or you fear her grief if she learns from someone else later, then it is better to inform her and try to appease her, especially given her health conditions and her inability to endure travels.