Your father's paternal uncle is regarded as an uncle to you too, and the husband's uncle is not a mahram for the wife, so it is not permissible for her to uncover her face in front of him, or for him to be alone with her, even if he is elderly, because of the general meaning of the evidence which enjoins hijab without making any differentiation between young and old. Allaah says (interpretation of the meaning):

"O Prophet! Tell your wives and your daughters and the women of the believers to draw their cloaks (veils) all over their bodies (i.e. screen themselves completely except the eyes or one eye to see the way). That will be better, that they should be known (as free respectable women) so as not to be annoyed. And Allaah is Ever Oft-Forgiving, Most Merciful"

[al-Ahzaab 33:59]

And the Prophet (peace and blessings of Allaah be upon him) said: "Beware of entering upon women." A man from among the Ansaar said: "O Messenger of Allaah, what about the in-law?" He said, "The in-law is death." Narrated by al-Bukhaari, 5232; Muslim, 2172.

Al-Layth ibn Sa'd said: The "in-law" is the husband's brother and other similar relatives of the husband, such as his cousin and so on.

The scholars of the Standing Committee issued a fatwa stating that it is not permissible for a woman to uncover in front of her husband's uncle even if he is elderly, because he is not one of her mahrams. Fataawa al-Lajnah al-Daa'imah, 17/424

But your wife may speak to your uncle and ask him how he is, without shaking hands or uncovering her face.

And Allaah knows best.