My best guess? Probably another primate. Bonobos and Chimpanzees seem like the ideal candidate to take over the husk of Human civilization the quickest. Another species might have a shot, but then it's a question of how many millions of years it's going to take for them to evolve and if they can survive the cataclysmic events that will no doubt hit Earth in the meantime.

If not primates, I would bet on one of the following species:

Corvidae - Extremely intelligent, highly adaptable, tool-users, social, pass down their knowledge to offspring.

Canis Familiaris - Highly social, apex predators, genetically diverse, spread throughout every corner of the world.

Loxodonta - Extremely intelligent, highly social, adaptable, builders and tool-users, long lifespans.