For years, I've been building and rebuilding an alternate Earth. The point of departure is 56 million years ago, when the hottest episode in the last 100 million years, the Paleocene-Eocene Thermal Maximum, lasted four to five times longer than it did in our timeline. Here is some of the backstory that is relevant to the question at hand, with some passages highlit in bold:
The evolutionary history of the primates, like virtually everyone else's, is long and extremely confusing. According to the molecular clock, the last common ancestor of all primates lived between 90 and 63 million years ago, yet we have found no fossils dating from that particular window. Apparently, the dry-nosed haplorhines (tarsiers, monkeys and apes) came first, with the wet-nosed strepsirrhines (lemurs, bushbabies and lorises) diverging from the haplorhines between 90 and 55 million years ago. There is conjecture that Plesiadapiformes were archaic primates, but there is doubt as to whether modern primates actually evolved from them. Two other groups, Adapiformes and Omomyoidea, throw a monkey wrench at the case because both groups appeared suddenly in the fossil record with no physical evidence of past transitions, and they were already diverse by the Eocene. So were those two groups already around to witness the Paleocene-Eocene Thermal Maximum, much less the fall of the dinosaur empire one million decades earlier?
Well, whatever the case, at 1750 species, the primates ended up being the most diverse of Great Lakes Earth's mammals, occupying the niches filled back home not only by our primates, but also by fruit-eating and predatory bats, sciuromorph rodents, carnivorans and even cetaceans.
The two superfamilies of carnivoran-like primates, the feliform-like terrailuroids and the caniform-like mixotherioids, have been identified as adapiform primates, which is amazing because back home, that particular group died out during the Miocene.
Here are some of those "carnivoran-like primates", all drawn from a commission by the DeviantArtist "AlienOffspring".
The dobarchu--an adapiform primate that looks like a cat that lives like an otter--and the dard, an adapiform primate that looks like a cat that lives like a badger.
The onza, a primate in a world sans Puma, and the pard, a primate in a world sans leopards and jaguars.
Two species of sphinxes, plains-running pursuit predatory primates in a world sans lions, cheetahs and scimitar-toothed cats.
A Bengal tiger used as a comparison for the two species of dubas, forest-exclusive primates that wrestle its prey like saber-toothed cats did in our timeline.
The lingbacker, an Arctic/Atlantic primate in a world sans rorqual whales.
The criticism here is that none of them look even remotely like primates, so what changes would be needed to better sell the audience that these are, indeed, primates in an alternate Earth?