It's probably fair to say that most scientific attention has been paid to the latter of those groups, but we do have a number of fossil prosimians from the Miocene. Madagascar, where most prosimian species are found today, was already an island, and had been since the time of the dinosaurs, but there are few known fossils of the relevant age there to tell us much about the early history of lemurs and their kin. Nonetheless, during the Miocene, prosimians were more widely spread than they are today.
Among those that seem to be fairly closely related to modern prosimians was Propotto. This was discovered in Kenya in 1967, and originally identified as a loris - lemur-like animals that live on the African and Asian mainland. However, the fossil consisted only of a bit of jaw, and just two years later, a more detailed analysis resulted in it being reclassified as a fruit bat. Despite regularly being cited as a rare example of a fossil fruit bat, and perhaps key to understanding the evolution of bats more generally, in 2018 it was moved back to the primates again, most likely as a relative of the aye-aye - which, if true, likely means that lemur-like animals colonised Madagascar at least twice, since the lemurs themselves were already there at this time.
Further north, in Asia, another group of early primates had clung on since much earlier times. These were the adapiforms, a third group of prosimians, distinct from both the lemurs and lorises. They were significant for having some features in common with early monkeys, although this is now thought to be a case of parallel evolution, likely due to a similar lifestyle. By the start of the Miocene, adapiforms had long vanished from the rest of the world, but they had clung on in southern Asia in the form of animals such as Siamoadapis from Thailand and Sivaladapis from Pakistan, India, and China. In most cases, all we have of these late survivors are teeth, although even these are enough to tell us that they probably ate a mixture of leaves and soft fruit, and that (assuming proportions similar to their earlier relatives with more complete fossils), the different species likely weighed between 0.5 and 4 kg (1 to 9 lbs).
The last of the adapiforms died out, leaving no descendants, around 8 million years ago during the Late Miocene, possibly because leaf-eating monkeys with more efficient digestive systems arrived in India at around that time and out-competed them.
These would have included colobus monkeys related to Mesopithecus, which lived across southern Eurasia from Italy to China shortly before the Miocene came to a close - although that seems to have eaten more hard seeds than its living relatives do. By this time, other types of monkeys, such as macaques, had already made their appearance, and the first baboons appear not much later, in the Early Pliocene. If we go much earlier than this, however, the picture becomes rather more confused.
Africa, where one might expect to find the most monkeys, seems to have been largely devoid of them during the Early Miocene. We know that they lived there earlier, so it's possible that they either lived in places that aren't conducive to forming fossils, or just in some of the many parts of the continent that have not been thoroughly surveyed for such things. They do, however, suddenly appear in the Middle Miocene, around 18 million years ago, in the form of Victoriapithecus from Kenya.
This has the great advantage that we have a relatively complete skull, something that isn't true of its earlier, pre-Miocene relatives. From this, we can tell that it had a relatively long snout, resembling baboons more than many other monkeys, and perhaps indicating that this was what most of the early Old World monkeys looked like. From the rest of its skeleton, we can tell that it spent a lot of time walking on the ground, which also implies that baboons, or at least macaques, may be more typical of the early monkey body type than we might suppose.
While Victoriapithecus could probably climb trees if it wanted to, and, at around 4 kg (9 lbs) was smaller than modern macaques, let alone baboons, it seems to have had an at least partially terrestrial lifestyle, and lived in open wooded areas, not dense forest. There is also some evidence that it lived in groups similar to those of modern macaques, and that the males were larger than the females. Nonetheless, the best guess, from its other features, is that it represents a now-extinct branch of monkeys that lived before the two living subfamilies diverged, making it no closer to modern macaques and baboons than to, say, colobus or proboscis monkeys.
The earliest monkeys in Asia, long before the colobus monkeys and macaques made their appearance, date from a little over 16 million years ago, likely having crossed over from Africa. These represent another extinct branch, so that it's not entirely clear where the direct ancestors of today's Old World monkeys were living at the time. In fact, they include the very first fossil primate to be described, way back in 1837.
This was Pliopithecus, first discovered in France, but now known to have lived across other parts of Europe, and possibly as far east as China and Mongolia. Whether or not that was really all one genus, a number of other monkeys belonging to the same family have since been described and the group seems to have been diverse. Most are known from fragmentary remains, but one, Epipliopithecus from Slovakia, has a relatively complete skeleton, giving us more of an idea of what it and its relatives were like.
For much of the late 20th century, Epipliopithecus had been thought to be a gibbon of some kind, or at least a close relative. That was largely due to the comparatively flat shape of the face, although there are some other points of resemblance. While we now know that it isn't, the fact that it does have features of both apes and Old World monkeys suggests that it may be descended from something that lived before those two groups had parted company.
They were also roughly gibbon-sized, with limb bones that suggest it could move about in the trees, but probably not swing from them as gibbons can, instead climbing in a more normal manner. While the rounded shape of the head would have resembled an ape more than a monkey, it did have a tail; we don't have a full set of bones from it, but the comparatively small size of the canal that would have carried its nerve and blood supply implies that it was probably quite a short one. They mostly seem to have eaten soft fruit, but there were several different species, and at least some probably ate some harder food from time to time. They died out around 7 million years ago, shortly before the end of the Miocene.
Prior to the appearance of Pliopithecus and its kin, there were no monkeys in Asia or Europe, and its clear that the ultimate ancestry of the group lies in Africa. But, even in the Early Miocene, that is not to say that monkeys did not exist elsewhere... because monkeys had already been in South America for some time.
In fact, there seems to have considerable diversification within the New World monkeys during the Miocene, with many of the modern groups first appearing in the second half of the epoch. While it's generally much harder to place the relatively small number of older species, we have a number of examples of monkeys from around 14 million years ago onwards that we can assign to living groups. For example, the leaf-eating Stirtonia from Colombia is thought to be an early relative of howler monkeys, while Neosaimiri from both Colombia and Peru, is an early squirrel monkey, albeit larger and likely less agile than the modern sort. Indeed, it is similar enough that it may not even be correct to assign it to a different genus than living squirrel monkeys.
A particularly intriguing question about Miocene New World monkeys concerns the origins of the marmosets and tamarins, unusually small monkeys now widespread in forested parts of the continent. They are highly specialised, with marmosets, in particular, feeding almost entirely on tree gum, and the group as a whole having re-evolved the claws that other monkeys have all lost. The first fossil marmoset was not described until the mid '80s. This was Micodon, and was considered to be a marmoset mainly on the grounds that it was too small to be anything else, since it didn't yet have the specialised teeth that we'd expect to see in the group.
While a few other fossils have, at various times, also been suggested as early members of the marmoset family, most have since been moved elsewhere. Nonetheless, we do still have a few, with Mohanamico, for instance, being somewhat closer to the modern forms. The fact that these specialisations seem to be missing in the earlier forms suggests that small size evolved first, since Micodon was already roughly the size of most modern species. Indeed, one analysis suggests that the original New World monkeys may have been marmoset-sized (roughly 400g or 14 oz.), and it's all the other families that have since grown to the larger size of the Old World sort.
Shortly, I'll be looking at what other mammals these Miocene New World monkeys shared their continent with. But before then, there's a rather obvious group of primates that I haven't mentioned yet...
[Photo by "ghedoghedo", from Wikimedia Commons.]