This is probably not the answer you were hoping for. There appears to be no mention of the origin of the tradition of Kumbha Mela in any of the Hindu scriptures.
From the research paper Making the Colonial State Work for You: The Modern Beginnings of the Ancient Kumbh Mela in Allahabad by Kama MacLean (p. 876):
Studies attempting to uncover the history and origins of the Kumbh Mela have strongly argued that these stories have been relatively recently applied to the festival (Bonazzoli 1977; Bhattacharya 1977; Dubey 1987, 2001; Sax 1987). R. B. Bhattacharya concludes that "the Puranic legend has been forcefully grafted on the Kumbha fair in order to show Puranic authority for it. Though the incident of amrita manthan churning for nectar has been stated in several Puranic works, 'the fall of amrita in four places' has not been stated in any of them" (1997 5).
Indeed, Francis Wilford's (1812) account of the ocean-churning story, published in the early nineteenth century, does not mention the drops of nectar either. Mention made of kumbh in various vedas (old Hindu sacred texts), puranas, and mahakavyas (great epic poems) has also been taken to refer to the mela, but this is taking considerable license in translation (Tripathi 1997, 275).
It is possible that the Magh Mahatmya and Matsya Purāṇa have references to "Kumbh Mela" but the paper claims (p. 881) the origin stories are most likely interpolated for other reasons:
That the authorization of the Kumbh Mela has been interpolated into books such as the Magh Mahatmya and Matsya Purana indicates the importance that the written word gained in Hindu practice, particularly in the colonial era. Richard King argues that this emphasis on textuality is partially the result of Western literary bias (1999, 102). The idea that all Hindu practice should find sanction in a holy book was one to which Orientalists clung, and they heaped scorn on those customs that did not meet this criteria and frequently used this as another reason as to why a particular religious rite was odious and aberrant and a further indication of the practitioner's ignorance - for example, the practice of sadhus proceeding naked at the Kumbh (Porter 1888). Consequently, practices recorded in texts were much more secure from colonial deconstruction; in this light, the interpolation of stories authorizing the Kumbh Mela was a wise move indeed. In Allahabad, as elsewhere, the regular observance of melas was predicated on effective advertising by priests and their agents, known to the British as pilgrim hunters, who regularly penetrated villages across the breadth of the country, alerting prospective pilgrims to a particularly auspicious forthcoming mela."
Also from the same paper (p. 888):
Religious Traditions, Innovation, and the Postmutiny Colonial State
I should note that I have not found any source, Indian or British, which clearly states that the Kumbh Mela was observed for the first time in Allahabad in the mid-nineteenth century; the reader will have noted that the absence of any reference to a Kumbh in Allahabad before 1860 throws doubt upon the ancientness of the Kumbh festival in that city. The argument of this paper, however, is that application of the Kumbh to the Magh Mela was necessarily surreptitious.
By appropriating a popular religious festival already well known – indeed, infamous – to the British and to their own city and by imbuing it with an immemorial past, the Pragwals sought not only to boost their business, but, more importantly, they attempted to create for themselves a sphere in which they could enjoy some autonomy from the increasingly repressive colonial state, which, after the mutiny and rebellion of 1857, had formally pledged not to interfere in traditional religious practices. As part of an accepted religious tradition, Allahabad's Kumbh Mela was unassailable in the eyes of British government, which had demonstrated its preference for tradition over innovation in the subject population.
And from p. 890:
Prayag is said to be contained in both Prayaga Ghat and Panchaganga Ghat, and Hardwar is said to be at Asi Ghat, where pilgrims are encouraged to imagine the Kumbh at mela times by the placement of "low wooden chauki platforms where the riverside vendors and priests are, stacked high with kumbhas, the fat clay waterpots, representing the kumbha of old which held the nectar of immortality" (Eck 1992, 140-46). This strategy clearly enjoys some success, which is evidenced by the crowds bathing in Varanasi in 2001 during the month of Magh, despite the fact that the widely lauded first Kumbh Mela of the millennium was being conducted a mere four hour train ride away ("Devotees Take Holy Dip," Times of India, 10 February 2001).
In this light, the application of the Kumbh Mela tradition to an existing religious festival – the Magh Mela – can be seen as an attempt by Pragwals to expand the fame of their tirtha.
The paper concludes with the following.
The Kumbh Mela has been successfully adapted to Allahabad and has arguably become the greatest mela in modern India, even eclipsing its Hardwar namesake.
While there are some religious and enchanted elements at work that emphasize Prayag's sanctity, other forces have promoted the Hardwar Kumbh Mela in Allahabad. By transcribing the Kumbh Mela festival onto the Magh Mela in the mid-nineteenth century, Pragwals were adapting their tirtha to suit the changing political and economic climate which otherwise may well have left them behind. In effect, the new Kumbh Mela became a vehicle for Pragwal, pilgrim, and sadhu aspirations. By carefully constructing a religious festival, they created for themselves a sphere in which they could enjoy some autonomy in the atmosphere of an increasingly repressive colonial state. Because these aspirations were articulated in the religious genre of the mela, the British conceded an element of sovereignty to them in recognition of the importance of religious freedom or, rather, in fear of the consequences if certain aspects of Indian autonomy within such a sanctified space were denied. Most important, the interpretation of Allahabad as a Kumbh site related it to a tradition which had a history well recognized by the British, so much so that there did not seem to be any invention at all. In the Allahabad Kumbh Mela lies an example of the way in which Indian actors appropriated and manipulated colonial discourse to their own ends: by extending from their own established customs and utilizing the tools of the British state, they found an arena of sovereignty.