Suppose everyone in the Shul remained awake the whole night of Shavuos. What do you do regarding Birchas HaTorah?
With regard to Birchas Hatorah, Mishna B'rura (47:28) brings a difference of opinion if one should recite Birchas Hatorah after a sleepless night, and therefore rules that one need not say the b'racha (ספק ברכות להקל). However, if possible one should hear it said by someone else and respond "amen", as is usually done Shavuos morning. If no one slept, though, it would seem this is not possible, and everyone should have in mind to be yotzei with the b'racha of ahava rabba before k'rias shema, and then say some p'sukim or learn some Torah immediately after shacharis.
However, if someone took a nap on erev Shavuos, then proceeded to stay awake the entire night, the Mishna B'rura brings the ruling of R' Akiva Eger, that he must recite Birchas Hatorah according to all opinions in the dispute cited above. That said, you will most always find someone who is obligated to say Birchas Hatorah.
Of course, as usual, for practical rulings, CYLOR.
According to Rav Ovadia Yosef (for sepharadim), you say morning berachot as normal starting from אלהי נשמה (and including birkat hatorah) at Alot HaShachar.
When this happened at my shul about 12 years ago, the rabbi simply sent someone up to say the b'rachos aloud on behalf of the congregation.