I worked on the show but don't recall Phillip ever clarifying who had had the stillborn. It was just a dried up fetus for Seth to find. It was also, as you point out, supposed to be somewhat mummified.
Also the two twin women in black, were not nuns but were supposed to be just whacky sisters.

It appears to be a stillborn baby but its apparently secretive burial suggests an illicit pregnancy, that is, out of wedlock, a scandal for the 1950s. The mystery is how come it hasn't completely decomposed and appears mummified.

Do you mean, that the films prop department went to the morgue or local hospital to acquire the fetus so that the film had the actual thing.
"I love the smell of napalm in the morning. The smell, you know, it smells like[sniffing],Victory."

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My suspicion is that the fetus belonged to the two nuns, who didn't seem to have any other role or purpose in the film. Other than to show there wasn't another goddamn person in town that wasn't mad as a loon.

SPOILERS
I think it was a real fetus, but regarding what you have pointed out - yes, the idea may just have been executed poorly (I believe the fetus would go limp after the first few days, but I could be wrong). You can note the fact that the kids said it 'smelled of fish' in the barn.
Anyway, aside from how the 'fetus' physically got there, was there some other significance associated with this element of the movie? Why did it look deformed?

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Seth and Kim come across the baby fetus as they're exploring
the barn
, because of the decomposition of the fetus they smell it's odor and dig it out. Someone has tried to hide it, but did a terrible job of it. Seth seems to think that its an angel of his friend Eban and decides to keep it under his bed. Sometimes he takes it out and talks to it, it seems to make him feel better.
"I love the smell of napalm in the morning. The smell, you know, it smells like[sniffing],Victory."