If a person has only one minute left before "Sof Zeman Keri'at Shema" – the final time for reading Shema in the morning – but he had not yet recited Birkot Ha'Torah that morning, should he read Shema before reciting Birkot Ha'Torah? Halacha requires reciting Birkot Ha'Torah in the morning before one learns Torah, and reciting the verses from the Torah that comprise the text of Shema certainly qualify as Torah learning. Therefore, under normal circumstances, one must recite Birkot Ha'Torah before reading Shema in the morning. However, in a situation where reciting Birkot Ha'Torah will result in forfeiting the Misva of reading Shema in the proper time, the question arises whether this Misva takes precedence over the obligation to recite Birkot Ha'Torah before learning Torah.
Hacham Ben Sion Abba Shaul (Jerusalem, 1924-1998) addresses this question in his work Or Le'sion, and he writes that in theory, reading Shema in the proper time takes precedence, and thus a person should skip Birkot Ha'Torah for the purpose of reading Shema in time. In practice, however, such a scenario will never actually happen. It is exceedingly difficult to pinpoint the exact moment of "Sof Seman Keri'at Shema," given the differences of opinion that exist, and the fact that a person's watch or clock may not be precise. And, reciting Birkot Ha'Torah takes just one minute. Therefore, a person will never be in a situation where he needs to read Shema before reciting Birkot Ha'Torah. If the final time for Shema is approaching, one should quickly recite Birkot Ha'Torah and then read Shema.
Summary: One must recite Birkot Ha'Torah before reading Shema. Theoretically, if reciting Birkot Ha'Torah would cause a person to miss the final time for Shema, then he should read Shema first, but in practice, this will never happen, because reciting Birkot Ha'Torah takes just a few moments, and it is all but impossible to pinpoint the precise minute by which Shema must be read.