Is Buddha Nature the original state, i.e. Awareness/ Consciousness without Existence?

OP: Is Buddha Nature Vedanta's opposite of Brahman i.e. Non-Existence, Consciousness, Bliss?

Firstly, what is Tathagatagarbha aka Buddha Nature? Is it the same as or opposite to Vedanta's Brahman?

According to this article:

Tathagatagarbha, or Tathagata-garbha, means "womb" (garbha) of Buddha (Tathagata). This refers to a Mahayana Buddhist doctrine that Buddha Nature is within all beings. Because this is so, all beings may realize enlightenment. Tathagatagarbha often is described as a seed, embryo or potentiality within each individual to be developed.

And it discusses further:

In the religions of the Buddha's day that were the forerunners of today's Hinduism, one of the central beliefs as (and is) the doctrine of atman. Atman means "breath" or "spirit," and it refers to a soul or individual essence of self. Another is the teaching of Brahman, which is understood as something like the absolute reality or the ground of being. In the several traditions of Hinduism, the precise relationship of atman to Brahman varies, but they could be understood as the small, individual self and the big, universal self.

However, the Buddha specifically rejected this teaching. The doctrine of anatman, which he articulated many times, is a direct refutation of atman.

Through the centuries many have accused the Tathagatagarbha doctrine of being an attempt to sneak an atman back into Buddhism by another name.

In this case, the potentiality or Buddha-seed within each being is compared to atman, and Buddha Nature -- which is sometimes identified with the dharmakaya -- is compared to Brahman.

You can find many Buddhist teachers speaking of small mind and big mind, or small self and big self. What they mean may not be exactly like the atman and Brahman of Vedanta, but it's common for people to understand them that way. Understanding Tathagatagarbha this way, however, would violate basic Buddhist teaching.

Another article here tries to point the origins of tathagatha-garbha or Buddha-nature to the Luminous Mind in the Pali Canon (which according to Theravada Buddhism, is not permanent or eternal, but is dependently arising - see this answer).

It comments in its conclusion:

In conclusion, when we try to interpret the thought of the 'tathagatagarbha', we should keep several points in mind:

1. The 'tathagatagarbha' symbolizes the potential for enlightenment (a principle) rather than a material "essence" of ultimate truth,

... Thus it is better to take the 'tathagatagarbha / Buddha nature' as representing "profound existence" derived from "true emptiness" rather than as a monistic self.

Buddha Nature is said to originate from the concept of the "luminous mind" in Pabhassara Sutta, which is anyway not eternal, not unconditioned and not impermanent (see this answer).

Nirguna Brahman is described in Advaita Vedanta as the substratum of all phenomena in Vivekachudamani 289 and as the material cause of the phenomenal universe in Aparokshanubhuti 45.

So, while Buddha Nature is simply the potential for enlightenment (a principle), it is not the material "essence" of ultimate truth, which Nirguna Brahman is.

So Buddha Nature and Nirguna Brahman are not the same. But they are also not opposite. They simply refer to different concepts. Like oranges and apples - not same, not opposite, just different.

OP: As awareness/ consciousness is always "on", no matter of what state it's in then surely awareness/ consciousness would still experience non-existence, akin to our lack of any experience in deep sleep.

Awareness/ consciousness in Advaita Vedanta is always "on". This is described by Bhagavad Gita 13.14 and Shvetashvatara Upanishad 6.11.

However, this is not the case in Buddhism.

The Buddha taught the following, from MN 38:

"Just as fire is classified simply by whatever requisite condition in dependence on which it burns — a fire that burns in dependence on wood is classified simply as a wood-fire, a fire that burns in dependence on wood-chips is classified simply as a wood-chip-fire; a fire that burns in dependence on grass is classified simply as a grass-fire; a fire that burns in dependence on cow-dung is classified simply as a cow-dung-fire; a fire that burns in dependence on chaff is classified simply as a chaff-fire; a fire that burns in dependence on rubbish is classified simply as a rubbish-fire — in the same way, consciousness is classified simply by the requisite condition in dependence on which it arises. Consciousness that arises in dependence on the eye & forms is classified simply as eye-consciousness. Consciousness that arises in dependence on the ear & sounds is classified simply as ear-consciousness. Consciousness that arises in dependence on the nose & aromas is classified simply as nose-consciousness. Consciousness that arises in dependence on the tongue & flavors is classified simply as tongue-consciousness. Consciousness that arises in dependence on the body & tactile sensations is classified simply as body-consciousness. Consciousness that arises in dependence on the intellect & ideas is classified simply as intellect-consciousness.

Think about it. How can the silent witness witness anything except through one of these media: eye, ear, nose, tongue, touch or mind? There was never a time, when there was consciousness being aware of something except through the eye, ear, nose, tongue, touch or mind. There is therefore no independent consciousness.

Consciousness is dependent on and conditioned upon these six media. It does not exist independently connecting all beings. The consciousness in every being may be of a similar type, but it's not the same consciousness.

For example, I can say that every candle has a similar flame, but it's not the exact same flame that appears on every candle. Each flame is different.

Awareness/ consciousness is not always "on" in Buddhism. It depends on the six sensory media.

Also useful, also from MN 38, in which the Buddha makes clear that it is not the SAME consciousness that wanders through one's life:

Then he went to the Blessed One and, on arrival, having bowed down to him, sat to one side. As he was sitting there, the Blessed One said to him, "Is it true, Sāti, that this pernicious view has arisen in you — 'As I understand the Dhamma taught by the Blessed One, it is just this consciousness that runs and wanders on, not another'?"

"Exactly so, lord. As I understand the Dhamma taught by the Blessed One, it is just this consciousness that runs and wanders on, not another."

"Which consciousness, Sāti, is that?"

"This speaker, this knower, lord, that is sensitive here & there to the ripening of good & evil actions."

"And to whom, worthless man, do you understand me to have taught the Dhamma like that? Haven't I, in many ways, said of dependently co-arisen consciousness, 'Apart from a requisite condition, there is no coming-into-play of consciousness'? But you, through your own poor grasp, not only slander us but also dig yourself up [by the root] and produce much demerit for yourself. That will lead to your long-term harm & suffering."

Good householder,

It's maybe good to find correction in regard of fundamental things, spoken on the Sublime Buddha's Dhamma, and the talk Freedom from Buddha Nature, starting with two quotes:

“What is the mind? The mind isn’t ‘is’ anything.”

“The mind is neither good nor evil, but it’s what knows good and knows evil. It’s what does good and does evil. And it’s what lets go of good and lets go of evil.”

...will help a lot, if wishing to get wrong perceptions corrected.

May many "Bodhisattas" have past of present collected strength of effort to go into it and gain more dhammic understanding.

[Note that this isn't given to keep one in the wheel of stacks, exchange and worldbinding trades of all kinds, but to escape from it for real liberation far off of that of "slave- and master-hood"]

Existence, Consciousness, Bliss is described as Brahman by Vedanta but surely that only applies when viewed through the veil of Maya and experienced as the universe. As awareness/ consciousness is always "on", no matter of what state it's in then surely awareness/ consciousness would still experience non-existence, akin to our lack of any experience in deep sleep.

Consciousness is not always on. Consciousness is conditional. All conditioned things arise , change and vanish.

Is Buddha Nature Vedanta's opposite of Brahman i.e. Non-Existence, Consciousness, Bliss? Knowing that existence arises from Buddha Nature, non-existence, is this why we start to grasp and cling to the dream of there being any existence at all? Not that there is nothing but that there are no-things, no names or forms, no self, just awareness/ consciousness?

Existence does not arise from Buddha's Nature which is the nature of unborn , uncreate and is unbecoming. Existence arises from Ignorance.

The Buddha state is beyond description, the Heart Sutra gives clear indications of this. The other clearest larger transmission I am aware of outside of the Heart Sutra for directly developing the Buddhic mindset is the Avatamsaka Sutra. Though this text seems unknown to the Tibetans it was widely transmitted in China, Korea and Japan. This text is also seen as having been influential in the formation of Chan/Zen.

The Hinaya/Theravedan path is indeed very concerned with mental precepts and, as taught by the Buddha, is perfectly fitting for those who work within the scope limitations. The Mahayana is the beginning of removing mental limitations by embracing the concept of Unity (not just my progression or my state or my happiness, but our progression, our happiness). The Vajrayana supercharges the path by engaging the science of the mind with transformative imagery, sounds, concentrations. The Avatamsaka Sutra, along with the Vimalakirti Sutra, begin teaching the student to enter the realization beyond name and form, to seed the mind stream with concentrations which are like a highway, or broadband network to the Buddha realization.

Special attention should be paid in the Avatamsaka Sutra to chapters 1-4, 6-12, 17-21, 27, 39. What is depicted here are not just stories, not just imagery, but direct mind-transmission from the Buddha realm into the reader. This is like a magical formula conveyed in a sequence of images/words. When combined with specific body concentrations the adept can learn how mudras and gentle positioning and tensations of the abdomen can activate energetic projectors where light originating from within extends and connects to all beings. The set of connections, which connect to all times and places, then dissolve into a time-less unity which is the entry into embodied pure lands and the Buddha sphere.

Much of what is here in buddhism.stackexchange is in dealing with the levels of limitations. Many lifetimes are eliminated as one uses the higher vehicles. Avatamsaka sutra, when practiced with the view and techniques of Vajrayana, does cross into an infra-zone with Dzogchen. Realization can be achieved in a single life time.