The Shulhan Aruch (Orah Haim 193:2) writes (listen to audio recording for precise citation) that a Zimun is required before Birkat Ha知azon even if only two people sat down together to eat, and only later were joined by a third person. Even though the first two people did not initially plan on eating together with the third individual, the three of them nevertheless form a Zimun, since in the end there are three people together at the meal. It goes without saying that if all three sat down together to eat, they may not subsequently split up and recite Birkat Ha知azon without a Zimun, since the Zimun obligation took effect once they sat down to eat.
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) cites the Gaon of Vilna (Rav Eliyahu of Vilna, 1720-1797) as ruling that a Zimun may be recited even if the first two people finished eating before they were joined by a third person. Since the intention of the first two people is to wait for the third person to finish eating so they can recite a Zimun together, they are still considered in the middle of the meal, even though they are no longer eating. Therefore, all three are considered as having joined together for a meal, such that a Zimun may be recited.
Summary: If two people begin eating together and are then later joined by a third person, they may recite a Zimun together, even though the first two finished eating before they were joined by the third, as long as their intention is to wait for him to finish eating before reciting Birkat Ha知azon.
