Backing up bowlturner, you've got a lot of problems here which together mean the answer is "no, this can't happen." You can get past them by authorial fiat, but if you want anything realistic it's not happening. So, the issues:
- Another species evolving to sapience in the near future is wildly unlikely. Dolphins, pigs, some apes, and (I think) whales are very smart animals. If anything was going to evolve to sapience, it would be these. Dolphins and whales live in water and they have no hands, so they're never going to get any sort of technology. Threat to us: zero. Apes have hands, upright posture, and the ability to make sound (and therefore conceivably language), so it's possible.
- Anything that does evolve to sapience, it's going to be a small number of individuals that make it at first. It will take a long time (hundreds to thousands of years at a minimum) for an entire species to attain sapience. We'll see it happening.
- This new species will be either starting with Neolithic technology (if it was an ape somewhere out in the jungle) or they'll be in a cage / zoo. Either way, not a threat.
You can get past it with sheer phlebotinum if you want -- "the aliens came and seeded the planet with a virus that made every non-human lifeform intelligent". Maybe they dump some nanotech that creates processing nodes in the brains and bodies of ants|parrots|monkeys|etc to give them self-awareness. Why did they do this?
If it was insects that became sapient, maybe we've got a problem. They massively outnumber us, they can hide in the walls, many of them are organized, they breed really fast, they have some ability to manipulate their environment, and it's hard to kill all of them. Anything roughly human size, there just aren't enough of them to be threatening and, again, they lack hands.
One thing: don't make the mistake of having super-size insects. Square-cube law means that they couldn't lift their own weight and, since they 'breathe' through spiracles instead of lungs, they couldn't get enough oxygen to survive.