I'm using some statistics out of William Boyd's 1956 printing of Genetics and the Races of Man
On skin pigmentation -- this is the earliest statement I've run across of the argument that the New World pigmentation cline is shallower than the Old World cline because of the relative recency of occupation (pp. 178-180):
The aborigines of the New World, though not by any means identical, agree in having on the whole considerable skin pigmentation. If pigmentation is adaptive, and conforms to climate, why are not the Eskimo and the inhabitants of Tierra del Fuego as light as Europeans? This looks like a considerable difficulty, but the solution is probably comparatively simple. The aborigines of the New World have not been here for more than about 25,000 years, or about 1000 generations. They are by origin Asiatic, and in Asia skin pigmentation is fairly heavy. Unless the selection of light skin as opposed to dark were fairly intense, the time elapsed has simply not been enough to allow for much adaptation to occur (12). As a matter of fact, the populations which might have been expected to become lighter, namely the Fuegans and the Eskimo, have probably had a shorter time in which to achieve this end than other American aborigines, for it is reasonable to suppose that the Fuegians did not reach their present home until long after their northern neighbors were well installed. And all students of the Eskimo agree in recognizing them as probably the most recent (aside of course from the whites) arrivals in America. It could well be that there has just not been enough time for selection to bleach the skins of the American aborigines.
Reference 12 is Haddon's Races of Man, which I have requested from the library.
I'm following up, because skin pigmentation is one of the traits most clearly subject to recent rapid selection. The new mutations that lighten skin tone in Europe and Asia are only partially shared between those populations. Many alleles are very common in one population, but nearly absent in the other. So far, the estimates of dates for these new variants are all within the last 20,000 years, but many remain undated. So we can't specify the level of pigmentation of people 15,000-20,000 years ago, yet, but it would have been substantially darker than those populations today.
Which leaves us with the same question, but from the opposite perspective. We now know that pigmentation evolved rapidly in Eurasia, the strong gradient of pigmentation having increased greatly within the last 20,000 years. We also know that the occupation of temperate South America began quite early, with people having been there longer than 10,000 years. So why did the New World end up with a more gradual cline -- darker pigmentation in the temperate and Arctic regions, lighter in the tropics than in the Old World? Was selection less intense? Can we attribute the difference to demography? Or chance?
Boyd next alluded to a demographic explanation -- low population density:
In any case, the pre-Columbian population was so sparse compared with that of Asia and India that on a statistical basis alone we should be justified in asserting that skin pigmentation conforms to climate.
Them's some tricky statistics.
We would of course today recognize that the sheer number of people is not especially relevant; much more powerful is the independent occurrence of a similar response in two long-separated populations. But Boyd was concerned with a different issue: Some had been claiming pigmentation as a neutral trait, making it more useful as a race marker:
This has been denied chiefly by those who were concerned to prove skin color a non-adaptive character, so that it might safely be used in the classification of races (12). Since the more up-to-date students of anthropology have given up the idea of relying on non-adaptive characters, or even believing that any such exist (13), there is no longer much dispute about the probable adaptive value of skin color (emphasis added).
Well, makes me glad to be an "up-to-date" student! There in fact has been an ongoing debate about "non-adaptive characters" as concerns the relationship of Pleistocene people. Many geneticists were surprised to discover the persistence of Neandertal genes, but in fact the skeletons of Upper Paleolithic Europeans clearly bear Neandertal traits. The debate for the last 30 years hasn't been chiefly about the presence of these traits, but instead about whether they were adaptive. Some argued that adaptive traits were not suitable evidence for a relationship, because they could emerge by parallelism in distinct populations.
Others observed that adaptive traits were more likely to be shared among populations linked by gene flow.
Now, of course, we have remaining unanswered questions about these shared traits. The shared traits are clearest between Upper Paleolithic Europeans and European Neandertals. We don't have genetic information yet telling us about the extent of Neandertal gene sharing with these early Europeans. Was it more than elsewhere? The traits would argue for it.
What about the Neandertal genes in populations far from Europe? One might expect Neandertal-like morphology to show up at some low level. Of course morphological features are polygenic, so that phenotypic resemblance falls much faster than genic identity. And Holocene populations have continued to evolve. Maybe early Asian skeletal remains like the Upper Cave skulls (ca. 11,000-20,000 years old) actually reflect that Neandertal heritage to a greater extent than recent samples.
Then there is the likelihood of other contributions, more local ones, to later populations.
Returning to the topic of pigmentation, many of us used to assume that the light skin of Europeans in part reflects Neandertal ancestry. That is, just as Boyd suggested, it would have taken a lot longer than 25,000 years to get the current strong cline of skin pigmentation in the Old World. If you could have longer, getting lighter pigmentation from earlier inhabitants of Europe, for example, you could explain a stronger cline with the same strength of selection.
I no longer think this is necessary. It's still possible that we got some pigmentation variants from Neandertals, but we haven't found any yet. And we've been looking. It does seem that Neandertals had some of their own pigmentation variants. Maybe we'll find many more of those, maybe not.