My friend remembers only scene, in which immortal people are after radiation. They are probably gray or silver (I think about color of their skin), they are bald completely and hidden in caves. They are reached by people from the top of the planet who have become immune to radiation or survived it. A scene similar to “Planet of the Apes”, because they meet these old people in the underground.
It isn’t “Time Machine” (1960), although Morlocks are some similar, but they aren’t bald. 
Possibly The Omega Man?
Maybe?:
“The Time Travelers” (1964)
https://www.imdb.com/title/tt0058659/?ref_=ttmi_ov_bk
Quote from the below Wikipedia plot:
”Pursued by hostile mutants, they seek refuge in a cave which they discover leads to an underground city – all that is left of civilization in a future devastated by nuclear war.”
Wikipedia plot summary:
On the movie’s IMDb page, in the Photos section there are pictures where they are bald, see two of those pictures below:
https://www.imdb.com/title/tt0058659/mediaviewer/rm2999831809/
Another guess:
”Beyond the Time Barrier” (1960)
Quote from the below Wikipedia plot:
”Despite this, the Captain sends Allison to an underground containment area, where a group of bald and violent mutants”
“Nuclear fallout suspended in the air damaged the Earth’s atmosphere”
IMDb page:
I will check this, however I am not sure. I will speak with my friend and I will check other "titles" too, because i have several another suggestions.
– wositelecHi, wositelec, I added two new suggestions, see them below.
– casspir