What is the significance of the phrase, "after Lot had separated from him"?
Seforno: HaSh-m did not want to inform Avraham this piece of news while Lot was there, for fear that it would go to the heads of Lot and his shepherds and would encourage them to rob their neighbors.
Rashi: As long as Avraham was in the presence of the Rasha, 1 HaSh-m declined to speak with him.
Chochmah u'Musar (Vol. 2, p. 370): Do not say that Lot was a Rasha. He disregarded his personal needs and abandoned his family to serve Avraham and learn Torah from him! Likewise, when it says about Shlomo that his wives veered his heart, it is not literally true. Rather, according to Lot's level, he did not decide properly, due to money. HaSh-m later showed him this - he lost all his property, and Avraham returned it. Refer to 14:12:1:2.
Why does the verse use the term "Va'Sh-m"? This always refers to HaSh-m and His Beis Din (see Rashi to Bereishis 19:24)!
Riva: Also here, HaSh-m's entourage wanted to kill Avraham due to the accompaniment of Lot, yet HaSh-m disagreed.
QUESTIONS ON RASHI
Rashi writes: "As long as the evildoer [Lot] was with Avraham, HaSh-m separated from speaking to him." This is difficult, because HaSh-m had already appeared to Avraham and said, "To your children I shall give this Land!" (12:7).
Moshav Zekenim: We can say [unlike Rashi] that Avraham was pained that Lot separated, so now HaSh-m consoled Avraham. Or, [Rashi] can say that earlier Lot was a Tzadik, and now he soured.
Riva: When Lot was with Avraham, HaSh-m spoke concisely, and now He elaborated.
Mizrachi: Avraham alone traversed the land until Shechem (to pray for Yaakov's children - Rashi to 12:6); that is when HaSh-m appeared to him. 1
Gur Aryeh #1: Before going down to Egypt, Lot was deemed insignificant to Avraham. Now that Lot had become rich and important in his own right, his proximity to Avraham was a deterrent to Avraham's prophecy.
Gur Aryeh #2: Verse 12:7 was an act of the gift of Eretz Yisrael, and is not called "speaking." This vision was to explain the extent of the gift, and HaSh-m waited until Lot parted ways.
Gur Aryeh disagrees; Lot was surely accompanying Avraham.