So "Hinduism" is a bit difficult to talk about because it is, to a degree, an artificial construction. What I mean by this is that Hinduism as a unified entity did not exist as a formal designation or deliberate religious tradition until late in its history. Rather, it was the amalgamation of religious traditions native to India which were considered "orthodox" by the Brahminical class and priesthood. However what was considered orthodox varied wildly over time. Many modern Hindus consider Buddhism and Jainism to be Hinduism, even though not two hundred years ago such a notion would be considered ridiculous or even offensive by Hindus. As such, we can see that the modern term Hindu is as much political as it is religious and is a bit loaded.

At this point we have to ask a question, prior to the British Raj, did "Hinduism" exist. I, along with scholars in the field, would say yes, with some caveats. Hinduism as a designation of religious identity dates to at least the Ghaznavids, where al-Biruni and other Islamic scholars identified the Indians as such as both a religious and geographical designation. A notion of solidarity between those who followed the Vedas and their Sramana opponents, however, existed as far back as the Sramanas themselves, long before the birth of Muhammad or even the Buddha and Mahāvīra!

But, the religion at the time of Buddha and Mahāvīra differed substantially from that that the Muslims would encounter. Ātman was a young and unorthodox concept, Samsara was rejected by some of the most orthodox Brahmins, Dharma was tied directly to sacrifices and caste, and ethics weren't even considered important beyond the rules of purity defined by keeping within a ritually pure community. Furthermore, Shiva was not a concept yet (Rudra was a distinct, marginally related god absorbed into a later cult, as it was already a designation) Visnu had little importance, Krisna legends were just barely a thing, and Shakti, Ganesha, Kali, and Durga were several centuries off. Indra, Agni, Ishvara, and Brahma seemed to be the gods with the most active cults and they little resembled their Puranic counterparts. So taking all this into account, what can we say about Hinduism in this period?

I would contest that since our modern, or even our medieval notions, of Hinduism are in question, we can argue that we are looking at the predecessor to Hinduism here, just like this religion had its own predecessor in the form of Indo-Iranian religion. This is often called Vedic religion. At the time of the Buddha, the Upanisads had only started composition about a century or two before and many major ones were still being written. What the Buddha and his Sramana contemporaries were responding to was both the Vedic religion and the emerging Hinduism from it. However, there was a very thin line between the practitioners of these Upanisadic traditions and emerging cults where they largely considered each other as part of the same tradition (and to a broad extent they were) while the Sramanas were radically distinct, rejecting the late Vedic and early Upanisadic notions of reality.

After the Buddha and Mahāvīra, a great many Sramanic innovations, like formalized monasticism, ethicized Karma, atomic theory, and ahimsa, would be absorbed into the emerging Hindu tradition along with the development of the cults of Shiva, Visnu, and Krisna and the further philosophical development of "orthodox" schools like Mimamsa, Yoga, Nyaya, Vedanta, and Samkhya. These philosophical movements were either contemporary, or more likely, slightly after the major Sramanic founders like the Buddha, Mahāvīra, and Makkhali Gosala and responded to their challenges in different ways.