Begins with future miners (digging up old rubbish tips looking to reclaim minerals from disposable nappies) finding a body, during the investigation, another body is discovered, then more, eventually millions of bodies are uncovered and the terrible truth is uncovered!
It is a short story by David Brin, Detritus Affected.
Future miners (archaeologists with industrial backing) digging old rubbish (and diapers):
Tried angling stories about disease germs, festering in the dump along with five billion ancient disposable diapers
Keoki pays the freight, testing rich veins of complex organics and heavy metals for our industrial sponsor. It's a long shot, but if the assay proves out, Fabrique Chang may bid to come mine Hyperion. One generation's junk can be the next's mother lode.
They find one skeleton
Zola found the bones down at South-22, a neat row of ribs sticking through a pile of dingy rags. At first we thought it was a pet, some large dog. On realizing they were human, we had no choice but to report it. We're digging in strata from A.D. 1958, after all. It might be somebody's long-missing great-uncle.
Today's big discovery—this time at South-31—four more sets of bones.
A million skeletons so far, with no end in sight. Reports come in from other cities. Of landfill boneyards in New York, Atlanta, Seattle . . . though none as extensive or dating so far back as Los Angeles. Perhaps that means it happened first and most profoundly here, in L.A.
But what? What happened here? Whence came the dead?
Ending explained in the spoiler block.
The archaeologists also receive threats telling them to stop digging. These are tracked down to people in care homes. The implication is that these are alien invaders who successfully killed and replaced the original humans back in 1958. There are other possibilities considered, but that seems the most plausible. However the narrator would prefer almost anything else.
While Detritus Affected strongly matches the majority of the details in the original question, as indicated in this answer, the plot element
reclaim ... from disposable nappies
suggests that the original poster might also be remembering details from Planet of the Sealies, where the diapers were actually of interest for
the genetic information they contain.