Most of the megafauna became truly extinct at the end of the Ice Age; they were not replaced by new animals, either evolved or immigrants, so today's fauna is a depauperate one. The Mountain Lion, or Puma, was present in Pleistocene times. We are not certain of its immediate ancestor; it may be that the Pliocene Felis lacustris is the ancestor of the modern puma, and may have been the ancestor of the American Cheetah, Miracinonyx, as well. There is still a lot to be discovered about those cats.
The sabertoothed cats were an entirely different clade, with no close living relatives.
The one Pleistocene fossil cat we do know something about is the American Lion, Panthera atrox, as it used to be named, and Panthera leo atrox as it is currently designated. A close relative of the African Lion and the European Cave Lion, it was an immense animal, and the apex predator of the North American Pleistocene.
Two good books, both rather out of date, but written for both a professional and a lay audience, which might help you are:
Kurten, Bjorn and Elaine Anderson, 1980, Pleistocene Mammals of North America, Columbia University Press.
Kurten, Bjorn, 1968, Pleistocene Mammals of Europe, Aldine Publishing Company
Rich