If Jed is Jethro's uncle, why is is Jethro's mother Pearl Jed's cousin? Is she Cousin Pearl or Aunt Pearl?

Jed and Pearl were first cousins, which makes Jed and Jethro first cousins once removed and Elly May and Jethro second cousins. I once pointed out that perhaps the characters started out as brother and sister as the E special said, and that somehow later on in the show, Paul Henning changed them to cousins, but Nick at Night is showing the first season and Jed called her "Cousin Pearl" from the very begining.
Jethro probably referred to Jed as
"Uncle Jed", because of the age difference and because Jed pretty much acted as de facto uncle to him. Same with Elly May and Pearl refering to each other as Aunt and Niece.
In those days and probably still today, many families use titles for relatives that reflect the emotional rather then blood relationship shared.
For example, everyone referred to Granny by that name, but she was only really the grandmother of Ellie May. I've seen shows where Granny called Jethro her grandson, but being that she was Jed's mother in law, she and Jethro weren't even blood related at all!

Erkie is right. In the South especially, it's common to refer to an older relative or even a family friend as "aunt" or "uncle", even when that person isn't really an aunt or uncle. And it's true that Jethro and Granny aren't really blood relations. Again, he probably calls her "Granny" as a term of endearment.

Even Jed had trouble explaining this one. In one episode when Jethro asked him if Granny married Mrs. Drysdale's father what kin that would make him, Jed tried to explain that Granny wasn't really his Granny. "Now my Paw and your Maw's Maw was brother and sister." He fumbles around some more with "so anyway; so that means", etc. before finally telling Jethro, "Anway, I think that makes you a great-nephew."
To which Jethro replies, "Thank ye, And I think you're a swell uncle."
"Thank ye."

The other responders are correct: I'm from East Tennessee, and we've had many people that we refer to as "Granny." Granny Jones, Granny Duggar, Maw Arwood, Mamaw Melton, Mamaw Hope, etc. A generation or two ago all adults were referred to as aunt and uncle. I remember one old lady (1900-1984) I was talking to once and she told how when she was a girl she was neighbors w/my grandfather about her same age. When his grandfather would come to visit, they all referred to him as "Grandpap". She knew he wasn't her grandfather, so she called him "Uncle Grandpap."