In truth modern science cannot not really answer this. Actually, it probably could, but… because of issues that are grounded in politics, racial debates, and the fears of indigenous people in the Americas (North & South), science has not been able to make its best attempt at solving the wide spread, and very hotly debated question that you ask.
Remains from a group of people known as the Clovis people have been found all over the American continents, as far as I know, more commonly in the Northern continent, than the south, however; the Southern continent, and Mexico (which is technically part of the North, but perhaps a more accurate description, one that is often used to reference the Mexican Region, would be Central America) has produced much much older anthropological finds, including artifacts and human remains from periods that predate the latest known period that the Clovis people lived in the Americas. If you read this article, you can get a good idea of the arguments raised in this subject.
The whole point to mentioning the topics that I did above, is to raise the question where did Native Americans come from, who were their ancestors? There is a wealth of data on this topic now that could help us answer these questions, but the “should be easy to collect data, has not been so easy to collect”. To tell what remains belonged to persons that were related, ancestrally, to the Native American Tribes people we must test the DNA of the remains, then test the DNA of modern Native Americans. Any attempt at doing this pisses someone off. Many tribes will not give consent to not only sample their DNA, but they have had legislation passed that prevents DNA testing of their ungraved ancestors, or of any peoples predating contemporary European settlement. In-fact many of the remains we have found, have had to be study rather quickly as legislation has forced scientist to rebury one of a kind precious remains. With that said, there has been some work done, with some useful data, but the conclusions produced from the data are arbitrary, and more data is needed to reinforce the claims made by Archeologists, and Anthropologists.
Link that demonstrates arguments about this subject
There is pretty good evidence that the Clovis people are very distant relatives of many of the Native American Tribes people, but what is hotly contested is the remains found that predate the Clovis people, DNA shows some similarities, but as stated previously, all DNA work that could be done, has not been done, and several chances to do DNA testing have slid right by us, and are no longer available due to the re-graving of remains that were never DNA tested when they were ungraved.
Currently it is theorized by some scientist, and this is becoming more widely accepted with the passing of each day, and not a lot of new data coming forth, several American Tribes are descendants of the Clovis people, this is easy to accept because DNA evidence suggests it. The group of people that predate the Clovis people show DNA similarities with the Clovis people, but they don’t show much evidence as being related to the North American Tribes, however, the key word is Much, there is little evidence, but not nearly enough to conclude concretely. Furthermore, upon the topic of scientific theory, it is hypothesized that the data shows exactly what you would expect to see when a new ethnic group is established. The latest ancestors don’t show a lot of DNA resemblance, but more recent ancestors show damning evidence that they are related to the new ethnic group, therefore; it is very likely that the reason we see less and less Native American resemblance as remains grown in age is simply because the remains grow less related with age, this is in fact how it works.
It's not as simple as I made it sound when I concluded the last paragraph, but I believe that what I have said, and so do many others, that what I have stated above is at-least partially true. It would be hard to say that it would take a specific number of years to form a new ethnic group, but when coming to a conclusion based solely on the data that is available, it looks like 24,000 years or more.