Praise be to Allah.
It is haraam for a man to shake hands with a non-mahram woman even if she is old. That is because of the general meaning of the evidence which states that it is forbidden.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him) said:
It is not permissible to shake hands with non-mahram women at all, whether they are young or old, and whether the man who is shaking hands is a young man or an old man, because of the danger of fitnah (temptation) that it poses to both parties. It was narrated in a saheeh hadeeth that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “I do not shake hands with women.” And ‘Aa’ishah (may Allaah be pleased with her) said: “The hand of the Messenger of Allaah (peace and blessings of Allaah be upon him) never touched the hand of any woman, and he used to accept their oath of allegiance by words only.”
It makes no difference whether one shakes hands with a barrier in between or with no barrier, because of the general meaning of the evidence, and so as to block the ways that may lead to fitnah. And Allaah is the Source of strength.
Fataawa Islamiyyah, 3/76
What some people believe, that the wife of a paternal or maternal uncle is not a stranger (non-mahram) is not correct, and there is no basis for that. Rather the wife of an uncle is a non-mahram. The question of a man shaking hands with a non-mahram woman has already been dealt with on this site; please see question no. 2459 and 21183
With regard to what the questioner says that not shaking hands will cause resentment and that this is customary among them, he should refrain from shaking hands with her, and should explain to her that he is not refraining from shaking hands because he despises her or thinks little of her, but rather because he is obeying Allaah and His Messenger. The Muslim has to obey the command of his Lord, and avoid that which He has forbidden, even if that goes against the people’s customs. But he should explain to the people that he is doing that out of obedience to Allaah, and he should be gentle and polite so that he can convince them to follow the sharee’ah and so that they will not think badly of him.
And Allaah knows best.