Is it permissible for my sister to travel to Makkah without a mahram, accompanied by my sick mother who needs her care, especially since my mother is accompanied by her nephew as a mahram for her?

Your sister is not permitted to travel with her cousin (maternal uncle's son) and his wife, because the cousin is not a mahram to her. The Prophet, peace and blessings be upon him, said: "A woman should not travel except with a mahram." The guardians of a woman must not be lenient regarding the mixing of their female relatives with those who are not mahrams to them, nor their seclusion with them, nor their travel with them, due to the great danger involved in such actions.