NOTE: both of these are only partial answers; the 1st may be against the Rashba and the 2nd is disputed
The simplest answer might be that the times of getting up and going to sleep are based on when non-Jews, who are exempt from Shema, wake up (after all, non-Jews do make up the vast majority of the human population). Rishonim (see Tos. 2b) ask a similar question regarding the time of night-Shema, if it depends on a poor person's meal time - when does this person say Shema? The Rashba (and other rishonim) writes that 'certainly we aren't dealing with a non-Jewish person', so presumably the Rashba thinks that wake up times of the Torah are dependent on those of Jews.
I haven't seen anyone who argues on this Rashba regarding the poor man, but I should note that the Gemara states David woke up at daybreak (daf 4a), and the Gemara specifically says there that 'all the kings of the west and east wake up at three hours', meaning, that this was the wake-up time for non-Jews kings.
When the Mishna says that one can read Keriyas Shema until 3 hours, it isn't clear whether this means 'the beginning of the third hour' (meaning, one can only read for two hours after sunrise), or 'the end of the third hour' (which would allow reading for three hours after sunrise). The Gemara asks this question regarding praying into the 4th hour (Berachos 26b-27) and answers that the mishnah means that it means the end of the 4th hour, so all four hours are included. Thus, we pasken (see Beis Yosef O.C. 58) that the same is true regarding Shema, that 'the 3rd hour' includes the entire 3rd hour.
However, regarding the actual wake-up time of princes, the Gemara states pretty clearly (see 3b) that they did so at the beginning of the 3rd hour, meaning only 2 hours after sunrise. Thus, Rabeinu Peretz (hagahos Semak Keriyas Shema 104) writes that the discrepancy exists specifically because the princes themselves need to say Keriyas Shema, which they do anyway. Similarly, the Bach (O.C. 58:2) writes that the princes needed time to get up and get dressed, meaning that the statement of the Mishna, דרך בני מלכים לעמוד בשלוש שעות, is that that's when they start their day, after they've gotten dressed and davened, etc. Thus, kings get up about an hour before the end of the zman according to R. Yehoshua.
The other time in the mishna, sunrise, also sounds like it was not a common wake-up time, but rather by which time most people were ready for prayer, as the mishna has cases of people being in the Mikvah from beforehand (Brachos 22b). However, this might not necessarily be the case, as the Gemara (daf 25b) notes that this mishna isn't necessarily referring to the latest time for shema, but merely the ideal time (kevasikin). Thus, it might be true that most people are not yet awake by sunrise, but because that's the beginning of the time that people rise, it's still זמן קימה. The Kehilas Yaakov therefore writes that the time of 'waking up' (as well as that of 'going to sleep') required for reading Shema must take this into consideration, and so the actual time intended by the Torah (he writes 'the Rabbis' but I assume he means the Torah) is actually slightly after wake-up time, enough to say Shema.
However, I should note that this position isn't universally accepted. Rabbeinu Simcha, quoted by the Hagahos Mordechai, (as well as other rishonim) believe based on the Gemara on daf 3b that the time for Keriyas Shema is actually only until, but not including the 3rd hour, and that they don't have that extra hour to get dressed and say Shema.