Must one fast for two weeks if one vowed to do so in the event of not reciting Surat Al-Baqarah daily, and then fell asleep during the recitation, which led to its completion after the beginning of the next day?

These two vows are "vows of obstinacy" (nadhral-lajaj), and the prevailing opinion is that one has the choice between fulfilling them or performing an expiation for an oath (kaffarat yameen). As for the first vow, one's obligation is discharged either by fulfilling it or by performing an expiation for an oath. Regarding the second vow, since the specified time for reciting Surat Al-Baqarah has passed (by the sunset of that day, not by the dawn of the following day), you are obligated either to fast for two weeks or to perform an expiation for an oath (feeding ten poor persons, or clothing them, or freeing a slave). If you are unable to do any of these, then you must fast for three days, and it is more precautionary that they be consecutive.