As we all know from the theme song, Old Uncle Joe was moving kind of slow at the Junction. "Petticoat Junction", that is, which was the nickname given to the water tower stop in Hooterville. The exact location of Hooterville was never revealed, but I have to figure it to be in the South, but not so far south that they never got snow during the winters.
Wherever Hooterville was, it's most likely that Joe Carson spent most of his life there.
Across the country in Los Angeles, Joey Carson was a baggy pants comic who owned his own nightclub. It was probably a wise business move to ensure he would always have a venue in which to perform. (The public's taste in comedy styles was changing. In the real world, more sophisticated acts like Nichols and May, Lenny Bruce, Mort Sahl, Shelley Berman, and Bob Newhart were coming to the fore. The same probably held true in Toobworld as well. There might even be a few fictional comics like that in various TV shows of the time.) As for where Joey Carson was originally from, it could have been from anywhere in the country, with Los Angeles being a magnet for entertainers to gravitate to.
There's only a fifteen year difference between Uncle Joe and Joey Carson, so a father-son relationship should be ruled out. (I prefer characters to be considered the same age as the actor playing the role, unless specifically stated otherwise.)
Brothers with the same name? It's happened before in Toobworld, with the two Arthur Dales in 'The X-Files' (who also had a sister - as well as a dog! - named Arthur Dales.)
And the fact that Joe Carson of Hooterville was known by everyone as "Uncle" Joe could be taken literally here - that Joey Carson was a nephew.
But I like the idea that they were cousins. Even as first cousins, there could be enough of a distance to their relationship which would make it understandable why Uncle Joe never mentioned his more famous kinfolk.
That they both bore the same name suggests that they were named after another relative of a few generations before. That Joe Carson could have been their grandfather, which would place him in the Civil War era.
And we have one to fit the bill! Joe Carson was an itinerant laborer when we met him in North Fork, a town in the New Mexico territory. He was wrongfully accused of murdering a cowboy for his horse and saddle, but he claimed that he bought them off the victim in Willow Springs. (As it turns out, the real murderer was the cousin of Lucas McCain's late wife.) Joe Carson wandered the West, finding work as a ranch hand, a bronco rider, whatever could make him a bit of money (usually for booze). But eventually he may have gone back home to the Hooterville area and started a family - with at least two sons to carry on the family name. (A close call with the hangman would be enough to sober up any man and make him take stock in his life.)
And somewhere along the way back east to the Hooterville area, never seen by the Trueniverse audience, Joe Carson may have done something to give him the reputation which would make both of his sons proud to name their boys after him.....
So there you have it - yet another Toobworld theory of relateeveety that can't be verified.....