As a Theravadan ive made the mistake of thinking Buddha Nature is just "everyone's ability to become awakened". There is apparently more to it then that but im in no way qualified to explain that as I have much to learn about it myself.
As for Theravada it is true as far as I have seen in my years of study and practice that "buddha nature" is not taught. However the Buddha says many times in the suttas that everyone has the ability to become awakened bases on their own effort and practice in this very life, so in that sense everyone has a "buddha nature".
Also it may be important to note the differences between arahants and buddhas between the traditions. In Theravada the only difference between a buddha and an arahant is that a buddha finds the path on their own when it has been lost, there is no difference between a buddha and an arahant in terms of wisdom/ability etc in Theravada, not so in Mahayana.
Consider this: Theravada posits Nirvana as a remote goal, while Mahayana posits Buddha-Nature as self-existing state of affairs. This is the key.
The reason Gautama Buddha declared Nirvana, is because he was modeling the Path after his own quest. Because he achieved his Enlightenment after having gone through numerous trainings and realizing Three Marks of Existence, Twelve Nidanas and Four Noble Truths, he saw that all students must go through similar experiences in order to achieve the same Liberation. Indeed, for someone without basic discipline, critical thinking skills, and capacity for self-reflection, attaining Buddha-Eye is outright impossible.
By the time of Mahayana though, because Buddhism became very popular and the essence of Teaching was somewhat lost in the noise of speculative philosophy and esoteric speculation, many new students would become obsessed with the idea of Nirvana, and instead of training the skilfull qualities of mind required for Enlightenment, would engage in fruitless search of Transcendental Realization. Out of compassion for future seekers, Bodhisattvas established the notion of Buddha Nature, openly declaring that Enlightenment is not something remote you have to attain, but rather one's innate nature to be recovered.
So Buddha-Nature is basically Nirvana or Enlightenment, except now you know you already have it, and only need to open your eyes, instead of running around the world seeking it. That said, you still have to work on dropping all attachments, deconstructing the ego, and becoming the master of your mind/emotions before you can take legal ownership of your rightful possession :)
That is because the main goal of Theravada practice is the state of Liberation (Arhathood), being free from first of the two veils (the veil of disturbing emotions). But reaching full Enlightenment (realizing Buddha-nature), when both veils are removed (the second veil is stiff ideas), becomes a topic only in Mahayana. That is because Bodhichitta, the great motivation for the sake of all beings, is an integral part of Buddha-nature and it is not stressed in Theravada tradition.
Theravada couldn't have all the possible teachings, so it has only some of them. The teaching about the Buddha nature probably didn't seem necessary at the time of Buddha.
It was rather obvious that practitioners could follow the eightfold path and come to awakening (Arahantship & Buddhahood). It was mentioned in Pali suttas, e.g. in Maha-parinibbana sutta (see the last talk of Buddha, with Subhadda the Hermit).
So the need to introduce "Buddha nature" developed later. As dialog develops, it often happens that more ideas need to be articulated to help people understand well. That's why Mahayana teachings developed, and they continue to develop nowadays.
For example, when I explain Dharma, I often use language, ideas and pictures which didn't exist in Buddha's times. I speak about objects compared to their photographs, and it makes easier to understand, for example, Diamond sutra, Two truths and the like.
Thus the concept of Buddha nature developed with time, maybe as a pinnacle of explanation of why everything. Indeed, limited people have limited views and goals. We do something because we want something, we are attached or repulsed, and so on. But why would Buddha act? If he isn't limited, free from wishes, from attachments and repulsions?
If you say Buddha acts for the sake of other beings, then please recall that beings are illusory, and suffering is illusory...
It's not easy to understand: why do anything, what could be the goal, if there is no attachment, no limits...
So as understanding of illusory nature of delusions developed - helping people to practise efficiently - the need to understand what is natural developed too.
If you are interested to understand Buddha nature in the context of the three main philosophical schools of Mahayana, and their application to Zen practice, see:
Master Chi Chern. Immaculate Self-Nature