Who propounded "All doesn't exist" in the Buddha's time?

There were many kinds of views in Buddha's time. This is discussed in Brahmajāla Sutta.

https://www.dhammatalks.org/suttas/DN/DN01.html

In relation to the translation "All Does Not Exist (Sabbaṃ Natthi)", SN 12.15 refers one duality of the world called "Non-Existence" ("Natthitā").

Similarly, the closest Pali term to the Western philosophical term 'nihilism' appears to be natthikavāda. It seems natthikavāda can cover many types of wrong views found in the Pali sutta, such as:

1. Sole reliance upon the dimension of nothingness, as taught by Alara Kalama in MN 26. As described in the suttas, the view of the dimension of nothingness is "There Is Nothing" ("Natthi Kiñcī").

2. The idea a "self" or "being" "dies", as found in Iti 49, SN 22.85 & DN 1 (ucchedavāda; annihilationism).

3. The idea a 'self' that used to exist no longer exists (as believed by Vacchagotta in SN 44.10).

4. The view another self/person causes oneself to suffering (SN 12.17).

5. Moral nihilism, as found in MN 60, MN 117 and DN 2 (Ajita Kesakambala's doctrine), as follows:

There are some contemplatives & brahmans who hold this doctrine, hold this view: 'There is nothing (natthi) given, nothing offered, nothing sacrificed. There is no (natthi) fruit or result of good or bad actions. There is no this world, no other world, no (natthi) mother, no father, no spontaneously arisen beings; no brahmans or contemplatives who, faring rightly and practicing rightly, proclaim this world and the other after having directly known and realized it for themselves.'

MN 60; MN 117

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When this was said, Ajita Kesakambalin said to me, 'Great king, there is nothing given, nothing offered, nothing sacrificed. There is no fruit or result of good or bad actions. There is no this world, no other world, no mother, no father, no spontaneously arisen beings; no brahmans or contemplatives who, faring rightly and practicing rightly, proclaim this world and the other after having directly known and realized it for themselves. A person is a composite of four primary elements. At death (kālaṃ karoti), the earth (in the body) returns to and merges with the (external) earth-substance. The fire returns to and merges with the external fire-substance. The liquid returns to and merges with the external liquid-substance. The wind returns to and merges with the external wind-substance. The sense-faculties scatter into space. Four men, with the bier as the fifth, carry the corpse. Its eulogies are sounded only as far as the charnel ground. The bones turn pigeon-colored. The offerings end in ashes. Generosity is taught by idiots. The words of those who speak of existence after death are false, empty chatter. With the break-up of the body, the wise and the foolish alike are annihilated (ucchijjanti), destroyed (vinassanti). They do not exist after death (paraṃ maraṇā).'

DN 2

At least three matters should be noted from the quotes above:

1. From MN 60, the verse does not necessarily refer to 'reincarnation' because the words 'world' & 'spontaneously arisen' are not necessarily something 'physical'. Instead, what is definitely certain is MN 60 is referring to both kamma & the results of kamma ('other worlds'). In other words, why 'Nihilism' is different to 'Buddhism' is because Nihilism does not believe in good & evil actions & results. For example, Nihilists are not grateful when they are taught the Dhamma. Instead, Nihilists believe: "there is nothing given, nothing offered, nothing sacrificed". This is Natthikavāda.

2. From DN 2, some of the views of Ajita Kesakambalin, such as a person is a composite of four primary elements, have similarities to what the Buddha taught (example, in SN 35.204). Therefore, not 100% of Ajita Kesakambalin's doctrine is different to Buddhism. What makes Ajita Kesakambalin's doctrine different to Buddhism is it is a doctrine of moral nihilism and also probably contains self-views.

3. Puthujjana Buddhists, obsessed with reincarnation or meta-physics, often say ucchedavāda (annihilationism) means the view of no life/no reincaration after death. However, annihilationism is simply the view a 'self' ('atta') or 'existent being' ('satta') is annihilated at 'death'. Therefore, ucchedavāda (annihilationism) does not mean 'no reincarnation' but, instead, refers to a 'self-view'. The Brahmajāla Sutta (DN 1) says:

There are some ascetics and brahmins who are annihilationists. They assert the annihilation, eradication, and obliteration of an existing being on seven grounds.

Santi, bhikkhave, eke samaṇabrāhmaṇā ucchedavādā sato sattassa ucchedaṃ vināsaṃ vibhavaṃ paññapenti sattahi vatthūhi.

And what are the seven grounds on which they rely?

Te ca bhonto samaṇabrāhmaṇā kimāgamma kimārabbha ucchedavādā sato sattassa ucchedaṃ vināsaṃ vibhavaṃ paññapenti sattahi vatthūhi?

There are some ascetics and brahmins who have this doctrine and view:

Idha, bhikkhave, ekacco samaṇo vā brāhmaṇo vā evaṃvādī hoti evaṃdiṭṭhi:

‘This self is physical, made up of the four primary elements, and produced by mother and father. Since it’s annihilated and destroyed when the body breaks up, and doesn’t exist after death, that’s how this self becomes rightly annihilated.’

‘yato kho, bho, ayaṃ attā rūpī cātumahābhūtiko mātāpettikasambhavo kāyassa bhedā ucchijjati vinassati, na hoti paraṃ maraṇā, ettāvatā kho, bho, ayaṃ attā sammā samucchinno hotī’ti.

That is how some assert the annihilation of an existing being.

Ittheke sato sattassa ucchedaṃ vināsaṃ vibhavaṃ paññapenti.

DN 1

4. SN 22.85 is a complex but excellent sutta, where a monk is admonished for having wrong view because he believes an Arahant is annihilated at death. This is wrong view because an Arahant is not "a being" ("satta") or a "self" ("atta") and thus does not experienced "death" ("marana"). Instead, SN 22.85 say at the termination of life (not "death"), the aggregates of an Arahant simply cease or end. SN 22.85 shows the term ucchedavāda (annihilationism) does not mean no reincarnation or no rebirth (but means the self-view that a self is annihilated).

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5. Lastly, while not explicitly related to the question, in later times, I think the Mahayana Heart Sutra is a perfect example of a doctrine: "All does not exist".

The doctrine 'Everything does not exist' is Nihilism and nothing to do with Buddhism. The Mahayana doctrine is 'Nothing really exists', which is a very different statement.

The word 'really' in this statement allows us to avoid extreme views. Things would exist in just the way they seem to, as dependent phenomena, but by reduction they would not. 'Nothing exists' would be an extreme view and the Buddha taught that we must avoid such views.

I've never come across anyone teaching that nothing exists.

This refers to the old Indian classification of astika vs nastica.

Roughly speaking, astika are the ones who believe in standard Hindu religious concepts (reincarnation of souls, karma, asceticism, liberation, the power of rituals, sacrifice, demons & spirits, other worlds etc.) while nastika are the ones rejecting that.

Buddhism is neither astika nor nastika, because instead of accepting or denying these religious ideas, it explains how consciousness develops and works, and then maps it back to some of these preexisting religious concepts, for convenience.

The most famous Nastika tradition at Buddha's times was called Ajivika.

The other obvious Nastika tradition is the naive materialism, the view of regular worldly people.

Perhaps it's a bit too frank, I think the OP asked an invalid question. A doctrine of everything doesn't exist cannot be uttered. It's like, a dead man cannot say "I'm dead", or you tell someone in the face "I'm not here". Making up such question is just an indication the questioner trapped by the trick of the mind - we called intellectual capacity.

This is the same trick for making up the phrase like the "groundless ground", the "egg-less egg", or the "valueless value"... the list can go on and on. You are simply making up another seemingly wisdom-infused phrase "non-existing existence". The Buddha has vivid and fun examples to dismiss, said, these people are talking about and debating on the "horns of a rabbit" and "hair of the tortoise". These things are mere constructs, the mind's game (戲論).

Let me make it more prominent: how can a question be made if all doesn't exist? What is to be discussed about since everything doesn't exist?

In ancient India, at Buddha's time, the study of Hetu-vidya and Catuṣkoṭi are prerequisites of establishing any doctrine. With all doesn't exist the whole proposition simply cannot be established, no learnt master would propound it.

Note:

I have great doubt on the OP's quoted Pali Sutta SN 12.15, the accuracy of the translated wording "everything exists" + "everything doesn't exist". Perhaps due to this misleading wording the OP asked the unnecessary question.

The corresponding Agama read as this (excerpt):

...尊者𨅖陀迦旃延...：「...云何正見？...」

佛告𨅖陀迦旃延：「世間有二種依，若有、若無，為取所觸；取所觸故，或依有、或依無。若無此取者，心境繫著使不取、不住、不計我苦生而生，苦滅而滅，於彼不疑、不惑，不由於他而自知，是名正見，是名如來所施設正見。所以者何？世間集如實正知見，若世間無者不有，世間滅如實正知見，若世間有者無有，是名離於二邊說於中道，所謂此有故彼有，此起故彼起，謂緣無明行...滅。」

佛說此經已，尊者𨅖陀迦旃延聞佛所說，不起諸漏，心得解脫，成阿羅漢。

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...Ven. Kātyāyana, "...what is right-view?..."

The Buddha told Kātyāyana, "There are two fixations in the world, if exist or if non-exist, due to attachment on contact. For attaching on contact, either fixate on exist, or fixate on non-exist. If one doesn't have any attachment, tames the mind made it not attach, not dwell, not involve the self. Suffering arisen as it arises, suffering ceased as it ceases. In it don't doubt, don't faze. Not depending on other but known it yourself, that is called right-view. This is called the right-view taught by the Tathagata. Why is that so? The aggregating of the world is known and seen as it is; if the world non-existed it is not existing now. The cessation of the world is known and seen as it is, if the world really existed it is not ceased existing now. This is called avoiding the two sides teaching the middle way. It is what called, this exists therefore that exists, this arises therefore that arises, called saṃskāra (activities) caused by ignorance... cessation."
... ... ... .. ... .

(the quoted italic text is corresponding to the OP quote.)

The Agama is much clear, concise and precise. Reading SN 12.15 ties a big knot in the head got lost in the bushes.