The Mishna Berura allows starting shnayim mikra for the upcoming week after davening Mincha on shabbos (285:7 - כיון שמתחילין במנחתא דשבתא לקרות פרשת שבוע הבא נחשב שוב הקורא כקורא עם הציבור וא"כ מה שכתב המחבר מיום ראשון ואילך לאו דוקא הוא). May one begin shnayim mikra for the upcoming week at the time he could have davened mincha or only after actually doing so?
The source for the Mishna Berura is from the Darkei Moshe (OC 285:1), who says:
...even from the time that they started reading the parsha by Mincha on Shabbat...
It seems from this that it would only be after the congregation has actually read the parsha by Mincha.
Note that the Darkei Moshe brings a dissenting view immediately after this, which says that one may not start at on Shabbat, only on Sunday, since they already read on that same day from the previous parsha.
Two sources suggest one can read from the time the congregation has prayed mincha, no matter if he has prayed with them.
The Mishna Brura (OC 285:7) writes (translation from the Ohr Olam MB edition)
Since we begin reading the next week's Torah portion (in shul) during Mincha on Shabbat, a person who reads from that time is considered to be reading together with the public (reading)
He doesn't note that the person himself has to be attending the reading.
Similarly, Shmirat Shabbat Khilchata (vol. 3, 42:58, p. 722) writes
A person can begin his reading for the coming week immediately following the Shabbat afternoon service, that is to say, after the congregation has begun the reading of the ensuing week's portion.