The first Mishnah in Berachoth quotes R' Eli'ezer's opinion that the time for reciting Shema' in the evening extends only until the end of the first shift (the Kohanim would serve in the Beith HaMikdash in three shifts throughout the night). If a particular set of Kohanim are on duty in the first shift, does that mean that, according to him, they can't say Shema' (or perhaps are exempt)?
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In the second mishnah of Brachos, there are two opinions as to what the latest time for Shema is: Rabbi Eliezer says sunrise, and Rabbi Yehoshua says it's three hours, because princes get up at three hours, so it can still be called "And when you get up" which is the time of the Shema in the morning.
The Rashba asked: If it is the practice for princes to get up at three hours, do they not fulfill their obligation to say Shema according to Rabbi Eliezer? The Shoshanim LeDavid (quoted by Tosafos Chadashim) answers that "When you get up" according to Rabbi Eliezer is when you personally get up. Therefore, since princes normally get up at three hours, they can still say Shema at that time.
We can apply this approach to your question. Apparently, since the kohanim on the watch are going to be consistently going to sleep later than that, they are allowed to say it later, according to Rabbi Eliezer.