Is it permissible to shake hands with the elderly wife of a paternal uncle if refraining from shaking hands causes aversion and hatred, and relatives have become accustomed to it?

It is absolutely forbidden for a man to shake hands with a non-mahram woman, whether she is young or old, and whether the man is young or old. This is due to the risk of fitna (temptation/strife). It has been narrated from the Prophet, peace and blessings be upon him, that he said: "Indeed, I do not shake hands with women." And Aisha, may Allah be pleased with her, said: "The hand of the Messenger of Allah, peace and blessings be upon him, never touched the hand of any woman at all. He would only take their oath of allegiance verbally." There is no difference between shaking hands with a barrier or without one. The wife of one's paternal uncle and the wife of one's maternal uncle are non-mahram. If refraining from shaking hands leads to animosity, the Muslim must still refrain and explain that he is doing so in obedience to Allah and His Messenger, doing so gently and kindly in his explanation.