If ten men ate bread together, they recite a Zimun before Birkat Ha’mazon and include Shem Hashem (Name of God). This means that the Mezamen (person leading the Zimun) summons the others, "Nebarech L’Elokenu She’achalnu Mi’shelo," and they respond, "Baruch Elokenu She’achalnu Mi’shelo U’be’tubo Hayinu." The Mezamen then repeats their response. If fewer than ten men eat together, then they recite the Zimun without "Shem Hashem" – that is, they omit the word "Elokenu."
If a group of ten men complete the Zimun and they then remember that they had forgotten to recite "Elokenu," they do not repeat the Zimun. The Shulhan Aruch (Orah Haim 192:2) writes that they have fulfilled their obligation despite omitting the Shem Hashem, and they should therefore simply continue with Birkat Ha’mazon, rather than repeating the Zimun with Shem Hashem.
If they realize their mistake at some point during the Zimun, should they then add Shem Hashem in the rest of the Zimun, or do they forfeit this opportunity once the Mezamen declares, "Nebarech She’achalnu Mi’shelo" without mentioning God’s Name?
The Halacha in this case depends on when precisely the people realized their mistake. If the Mezamen omits Shem Hashem in his introductory declaration of "Nebarech…" and the others immediately inform him of his mistake, then he should repeat "Nebarech" and add "L’Elokenu." If the Mezamen did not correct his mistake, but the others responded to him by declaring, "Baruch Elokenu," with God’s Name, then he may likewise respond, "Baruch Elokenu." Since they recited "Elokenu" in their response, he, too, should add "Elokenu" when he repeats their response. However, if they responded incorrectly, by reciting, "Baruch She’achalnu," without "Elokenu," then he, too, must repeat "Baruch She’achalnu," without God’s Name. Once the others respond without "Elokenu," he can no longer recite "Elokenu" when he repeats their response.
This is the ruling of several Halachic authorities, including the Ben Ish Hai (Shana Rishona, Korah, 2), the Kaf Ha’haim (Laws of Zimun, 16), and Rabbi Moshe Halevi (Birkat Hashem, vol. 2, p. 496).
Summary: A group of ten men who ate together recite a Zimun with God’s Name ("Elokenu"). If the one leading the Zimun begins without reciting "Elokenu" and the others correct him, he should repeat the first line with "Elokenu." If he did not repeat it correctly but the others responded with "Elokenu," then he should repeat their response with "Elokenu." However, if the others respond without "Elokenu," then even though they realize their mistake afterward, they should simply continue with Birkat Ha’mazon without repeating the Zimun.
