If a group of three or more people ate a meal together with bread, such that they are required to recite a Zimun before Birkat Ha’mazon, but they mistakenly recited Birkat Ha’mazon without first reciting a Zimun, they can no longer recite a Zimun. The Zimun can be recited only before Birkat Ha’mazon, and so if the group mistakenly recited Birkat Ha’mazon before reciting the Zimun, they cannot then recite the Zimun afterward.
This applies as well if three people ate together and two out of the three recited Birkat Ha’mazon without first reciting a Zimun. This group can no longer recite a Zimun.
However, if only one out of the three made the mistake of reciting Birkat Ha’mazon without a Zimun, but the other two had not yet recited Birkat Ha’mazon, the three of them can still recite a Zimun before the other two recite Birkat Ha’mazon. In such a case, those two fulfill the Misva of Zimun. The third does not fulfill the Misva – since he had already recited Birkat Ha’mazon – but he can nevertheless join the other two to enable them to fulfill this Misva, since he had been part of the meal. He recites the regular text of "Baruch She’achalnu Mi’shelo" just as he would if he had not first recited Birkat Ha’mazon.
The Bi’ur Halacha (essays accompanying the Mishna Berura by the Hafetz Haim) rules that this Halacha also applies in the case of ten people who ate together, three of whom mistakenly recited Birkat Ha’mazon without a Zimun. In such a case, the Bi’ur Halacha writes, those three can join the other seven for a Zimun such that the Zimun may be recited with the Name of Hashem ("Nebarech Elokenu She’achalnu Mi’shelo"). Since these three had been part of the group, they may join the others to form a Minyan for a Zimun even though they already recited Birkat Ha’mazon. Although the Hid"a (Rav Haim Yosef David Azulai, 1724-1806), in Birkeh Yosef, is uncertain about the Halacha in such a case, this is the ruling of the Bi’ur Halacha, as well as other authorities (including the Panim Me’irot and Kaf Ha’haim).
Summary: If a group mistakenly recited Birkat Ha’mazon without reciting a Zimun, a Zimun cannot then be recited afterward. However, if only one out of the three recited Birkat Ha’mazon without a Zimun, he can still join the other two to enable a Zimun (though he is not credited with the Misva of Zimun). Likewise, if ten people ate together, and three mistakenly recited Birkat Ha’mazon without a Zimun, they can join the other seven to allow for a Zimun with Hashem’s Name ("Nebarech Elokenu She’achalnu Mi’shelo").
