Is a mother sinful if she informs her family about her daughter's diagnosis with tuberculosis of the bones and spine, despite her daughter's request not to tell anyone, claiming the disease is not contagious, and the mother's fear that it might be contagious and that they might visit her?

There is no harm in asking specialists about a disease and whether it is contagious or not, without identifying the infected person. If it is proven to be contagious, then the infected person must be avoided. The disease has no effect except by the permission of Allah, and there is no sin upon the one from whom the disease transferred to another without any fault of his own.