Can his wife uncover in front of his elderly uncle?

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.