The Hogben series by Henry Kuttner and C. L. Moore. Here is a video ad by Neil Gaiman for The Hogben Chronicles. Here is the list of stories from the ISFDB:
- "The Old Army Game", first published in Thrilling Adventures, November 1941. (The ISFDB notes that this one "features a hillbilly character named 'Hogben', but the story is not SF and is not otherwise related to the subsequent series of Hogben stories.")
- "Exit the Professor", first published in Thrilling Wonder Stories, October 1947, available at the Internet Archive.
- "Pile of Trouble", first published in Thrilling Wonder Stories, April 1948, available at the Internet Archive.
- "See You Later", first published in Thrilling Wonder Stories, June 1949, available at the Internet Archive.
- "Cold War", first published in Thrilling Wonder Stories, October 1949.
Not counting "The Old Army Game", the Hogben stories were published between 1947 and 1949 in Thrilling Wonder Stories. They do indeed featured a family of mutant hillbillies with supernatural powers. Quoting from "Exit the Professor":
He pulled out a notebook and started looking through it.
"I'm a—a scientist," he said. "Our foundation is studying eugenics, and we've got some reports about you. They sound unbelievable. One of our men has a theory that natural mutations can remain undetected in undeveloped cultural regions, and—" He slowed down and stared at Uncle Les. "Can you really fly?" he asked.
Well, we don't like to talk about that. The preacher gave us a good dressing-down once. Uncle Les had got likkered up and went sailing over the ridges, scaring a couple of bear hunters outa their senses. And it ain't in the Good Book that men should fly, neither. Uncle Les generally does it only on the sly, when nobody's watching.
Lemuel is introduced in the opening paragraph of "Pile of Trouble":
We call Lemuel "Gimpy," on account of he had three legs. After he got his growth, about the time they fit the War Between the States, he was willing to keep his extra leg sort of tucked up behind him inside his britches, where it would be out of sight and people wouldn't talk. Course it made him look a little like one of them camel critters, but then Lemuel never was vain. It was lucky he was doublejointed, though, or he might of got cramps from keeping his leg tucked up thataway.
From skimming the story, it looks like he is always just plain "Lemuel", not "Uncle Lemuel" or "Cousin Lemuel". There is however an Uncle Les, mentioned in the first quotation.