The Gemorah Bovo Kama (Merubah) by the ten 'Takonos' of Ezra, Daf 82a, says that the Neviim between them were 'mesaken' to read on Shabbos, Monday, and Thursday. I would assume that they read from what we call Chumash. And the Rambam in Hilchos Tefila Perek 12 Halacha 1 starts the laws of Krias HaTorah by saying: "Moshe Rabbeinu was 'me-sa-ken' for the yiden to read in [the] Torah publicly, Shabbos, Monday, Thursday," not to go three days without hearing Torah. Ezra built on this 'takono' and added Mincha Shabbos to the list.
It sounds like what we do, our reading of the Torah, started from the times in the 'Midbar.' Pre-Seenai. If that is true, I wonder how it was done if there was no 'Sefer Torah,' (although neither the Gemora nor the Rambam uses the words 'Sefer Torah'). What else could it have been? Parchment with Sefer Breishis? Selected Parshiyos? Plain Halachos? According to Rashi, Ramban, the writing of the Torah didn't start at least until Har Seenai. They must have read something. And they read throughout the 40 years before the Sefer Torah was finished .
Yasher Koach.
Moshe
A number of Acharonim ask your question (see, for example, Toras Chayim).
First of all, the Rambam's view is not accepted by all. The Gemara at the end of Megilah mentions Takanos of Moshe (to discuss the Chag during the Chag), but does not mention that he said to read the Torah three times a week. When Rav Nissim Gaon brings a list of Moshe's Takanos (Shabbos 30a), he does not mention anything about the weekly Torah readings. If so, the Gemara may simply mean that early Nevi'im, basing themselves on the verses about Refidim, instituted that the Torah be read three times a week. This is, in fact, the approach that the Toras Chayim (and Torah Temimah Shemos 15:34) takes to your question.
But even according to the Rambam, there is no need to assume that Moshe made the Takanah as soon as the story of Refidim occurred. He may have made the Takanah when he handed the completed Torah to Yisrael. (In fact, the Gemara does not say that "Miyad..." the Nevi'im made a Takanah....)
Best wishes,
Mordecai