It is really tricky to not have a sense of time. Any predator or prey has an intimate understanding of time, because time is literally a matter of life and death. But is there another way?
Perhaps we can borrow from the computer world. Computer systems often treat time very differently. In your question you talk about a species that "never [learned] the concept of time." You used a definite article in that sentence: "the concept of time." As though there is only one. It turns out that time is actually quite complicated, and if you delve into the philosophy, you find there are many understandings of time. However, for this answer, I'm going to assume you refer to the continuous time that we use in equations of motion in science. It's the only one which is ubiquitous enough to earn that definite article.
It's also a meaning of time that computers tend to not have. When we design most algorithms, we simply don't have that concept of time. An algorithm runs as fast or slow as it needs to, and when it's done, it's done.
So how do computers react in a world which appears to be ruled by our concept of time? One of the main approaches is to use what are known as "interrupts." Basically, these are specialized signals which, when triggered, interrupt whatever processing was being done and force the processor to run another bit of code. For example, when you press a key, it sends an interrupt to the computer which basically says "a key got pressed. You should read which key is being pressed right now, because they're going to let go soon, and then no keys will be pressed for you to read."
Interrupts create an interesting pattern. They don't have the sense of time that we have, used in Newtonian calculus. What they have instead is a concept of causality and "happens before" relationships. Believe it or not, you can have the concept of causality without a sense of time as we know it. Instead, you just have a series of well ordered events.
What I like about this pattern is that said species is not immediately turned into chow by a predator that has a sense of time as we know it. This species could have interrupts set up in their mind such that when the predator acts, they react. They may be unaware of the time flowing, but they can memorize a set of reactions which keep them safe from a sabertooth tiger. It might even be very aggressive. They may have a trigger which tells them to jab the tiger in the eye. They could do this without a sense of time, so long as they didn't have to solve any calculus equations to predict where the tiger's eye is. Accordingly, they would live in an environment that permits this. They would live in an environment full of complex traps, such that once a tiger falls into a trap, it remains stationary long enough for the members of our species to figure out how to dispose of it without having to do any complicated timing.
What they would be very bad at is balancing acts. Without a sense of time, tasks like balancing a broomstick on their hand would be very very difficult. Bang-bang controllers, which are independent of the flow of time, are not very good at this task. However, they could actually survive. And, in the ideal situations, maybe even thrive.