Let’s say that the bulk of the big mammals hold on a hundred centuries longer than they did in our universe. They survived the first group of apes to walk into the new world, but then some others arrive, this time across the Atlantic from Europe.
Personally, I’d guess that if many weren’t tragically hunted to extinction, they would still face massive population declines. Mammoths and mastodons would be under siege for their ivory, and even if federal protections stepped in at the last minute, it would take them a very long time to recover. Large predators like sabertooths, American lions, dire wolves, and short-faced bears would be persecuted to protect people and livestock, not to mention hunted for sport. Since in our timeline we almost killed off all the bison in order to kill off the native people of the plains, perhaps these alternate universe ones would attempt to inflict genocide on not only them but also camels, tapirs, ground sloths, musk oxen, and the list goes on.