If I start the Shema before the zman but I finish the 1st 3 paragraphs after the alloted time, what is the halacha?
There is a machlokes haposkim what portion of Shma is min hatorah (Aruch hashulchan 58:16: 1. First pasuk 2. first parsha 3. First 2 parshios 4. All 3 parshios
Thus, if the sof zman krias shma arrived during Shma, at least you fulfilled the mitzva min hatorah according to so opinions.
There is a fascinating Aruch Hashulchan 110:5 based on Tosfos (who writes that Bilaam only started to curse Bnai Yisroel when Hashem was angry but concluded the curse afterwards) who holds that if one starts during the zman one can finish after the zman. His opinion is probably limited to Tefilah whose time limit is derabanan and does not apply to Shma whose time is min hatorah, but his proof from Bilaam might imply otherwise.
If the question was regarding fulfilling one's obligation to recite Shema' in the morning, Qizur Shulhan 'Arukh - Yalqut Yosef (Orah Hayim 58:2) states (my translation):
Within Zeman Qeri'at Shema' one must read all three paragraphs and not just the first.