I imagine this isn't quite what you had in mind when you say "oral tradition", but ancient texts such as the Bible are generally the result of writing down or merging traditiong that themselves may have been transmitted orally for centuries before being committed to paper. Therefore, when historical-critically analyzing a text like the Bible, you're usually both analyzing the time and place of the underlying traditions, and the time and place of the editor who compiled the received text. In the particular case of the Torah, the commonly-held Documentary Hypothesis posits four main sources: The Yahwist J, Elohimist E, Deuteronomist D and Priestly Source P, with P often being considered the "editor" of the final text (or a prototype of it). J and E are typically thought to go back several hundred years before they were first written down.

One of the most extreme cases of discrepancies between origin and comission to written medium are found in my own field of primary interest, viz., the Gathas, the seventeen hymns of Zarathustra. They are written in Avestan, a very archaic Iranian language which in its most archaic form (conventionally called Gathic) is reasonably mutually intelligible with Old Indic (the Sanskrit of the Rgveda, another example of a long-running oral tradition put to paper). Comparing the Gathas to the so-called Younger Avestan corpus (Yashts, or hymns to divinities, the rest of the Yasna, or liturgy, as well as the Videvdad, a Late Antique compilation of various writings) and sketching out a timeline based on linguistic and theological developments (e.g. the use of Persian loanwords starting around the Achaemenid period, or the incorporation of characteristics of Near Eastern deities), it is possible to arrive at a broad date for the Gathas as having been composed by a single author (viz., Zarathustra) between about 1500 BC and 1000 BC, with 1300 BC being the most commonly suggested approximate date. The oldest layers of the Rgveda, by comparison, are thought to have been composed between about 1500 and 2000 BC. The dating of the latter is bounded by linguistic analysis dating the split between the Indic and Iranian languages to about 2200-2000 BC.

Now, Iranian tradition places a very strong premium on oral transmission, and it is typically thought that the Gathas and the rest of the Avestan corpus was written down no earlier than the Sasanian era, that is, between about 200 and 650 AD, for various reasons including the emergence of genres of Persian literature and the peculiarities of the Aramaic-derived writing system employed for Avestan. The Gathas are in a sense somewhat easier to date than other material, because they are so archaic in their language, appear relatively intact, and because they were composed by a single author. By comparison, only very broad ranges can be assigned to the Yashts, as these somewhat less holy texts were probably subject to modifications, insertions, etc over a long period of time. Usually it is thought that the Yashts were mostly codified during or shortly before the Achaemenid period, with the oldest ones (such as the Yasht to Mithra) having a core tradition going back to perhaps 1000 BC.

Concerning not only the time but the place, the Gathas are above all recognizable in the very sparse, pastoral lifestyle they depict, with only vaguely defined social hierarchies and little material culture. This has lead authors in recent years to place Zarathustra's homeland somewhere around today's Uzbekistan, or perhaps as far north as Kazakhstan. The Yashts and the archaic first chapter of the Videvdad are notable for featuring geographical toponyms that only seem to include Central Asia, and none of West Iran, which is one thing that suggests that their composition mostly pre-dated the Achaemenid period - indeed, in the Sasanian era, some Central Asian placenames would be re-interpreted as referring to Mesopotamia and West Iran!

So, while I understand this may not be exactly what you had in mind with respect to "oral traditions", I hope it still gave you some insight into how the method can be applied to them.