See the following related discussion:
Question: I have questions about the Kri’at Hatorah (Torah reading = laining) of Mincha of Shabbat regarding cases (common in the army) where we will not have access to a sefer Torah at the time we can daven Mincha. Can we read it in the morning after Mussaf? If we davened without Kri’at Hatorah and then we got access to a sefer Torah, can we lain at that point?
Answer: The poskim discuss whether people who missed Kri’at Hatorah at Shacharit during the week can make it up at Mincha. On one hand, the logic of the Kri’at Hatorah (not going three days without public Torah learning- Bava Kama 82a) applies all day. On the other hand, once the Rabbis set a time to do this, perhaps that cannot be changed, as the Rambam (Tefilla 12:1) seems to posit. In practice, most Ashkenazim allow the Mincha make-up (Mishna Berura 135:1; this was Rav Soloveitchik’s weekly practice). Sephardim are less likely to accept this practice (see Yosef Ometz 27; Yabia Omer IV, Orach Chayim 17).
It is much more problematic to do the Kri’at Hatorah of Shabbat Mincha earlier. The gemara (Bava Kama 82a) says that this reading was instituted because of yoshvei keranot. Rashi explains that these are people who do not hear Kri’at Hatorah on Monday and Thursday, who get to hear an additional reading on Shabbat. The Shita Mekubetzet says that since people are liable to get drunk during the day, Kri’at Hatorah was instituted to encourage them to go to shul. According to the Shita Mekubetzet, the time of day is an important part of the institution. Even according to Rashi, claims that Tzitz Eliezer (X, 20), it appears that Kri’at Hatorah of Mincha was instituted as a distinct laining in terms of content and timing, and therefore, when the gemara (Bava Kama 82a) says “at Mincha,” it means it. The Tzitz Eliezer also cites Kabbalistic reasons for having the Kri’at Hatorah (and U’va L’tziyon) at the time of Mincha. Based on the above, even if a community will not have the opportunity to lain at Mincha, they may not do so in the morning.
Doing Kri’at Hatorah in the half hour between chatzot (astronomical/halachic midday) and the time in our calendars for Mincha is far less problematic. There are several indications that, fundamentally, the time for Mincha is chatzot, just that we wait half-an-hour to make sure chatzot has passed (see Tosafot, Nida 63b; Magen Avraham 458:1). We know of no proofs that this precaution applies to Kri’at Hatorah, although the Tzitz Eliezer (ibid.) assumes it follows the same rules as Mincha. Also, the Mishna Berura (233:2) raises the serious possibility that after the fact, if one davened Mincha after chatzot he fulfilled his mitzva. Therefore, if the latest that a group can lain is during this half-hour, they may do so.
Regarding Kri’at Hatorah after davening Mincha, the Eshel Avraham (Butchatch, 292) is uncertain about the matter. Yaskil Avdi (VIII, OC 38) assumes that people who have already davened may lain afterward. During the week, it is very accepted (see Yalkut Yosef, OC 135:(6)) to do Kri’at Hatorah after davening when it was not possible to do it in its place (e.g., if people daven in a mourner’s home without a sefer Torah). Our case is a little worse in that Kri’at Hatorah should be before Mincha, as opposed to at Shacharit, where Kri’at Hatorah is anyway after Shemoneh Esrei (the main part of tefilla). However, there are strong indications that, fundamentally, Kri’at Hatorah is a separate obligation from tefilla, even though it is preferably attached to it. One such source is the Terumat Hadeshen (15; see Shulchan Aruch, OC 55:2) who says that if there was a minyan to start Chazarat Hashatz and then people left, the remaining group can continue all parts of tefilla that are connected to Shemoneh Esrei (including Kaddish Shalem). However, he says, they can not lain because it is not intrinsically connected to tefilla. Therefore, we conclude that a minyan of people can lain after Mincha if a sefer Torah becomes available then.