According to general halacha, one may not recite the berachot of keriyat shema after the 4th halachic hour. Ashkenazim can do so if they were withheld from reciting them in their proper time for reasons outside one's control (ones)1, and sefardim can do so without shem umalchut2.
Another halacha is that one should ideally say the important shemona esre before the 4th hour is over (i.e. be finished saying it by then)3.
Imagine the scenario that someone realises they only have 8 minutes left until the end of the 4th halachic hour, which is either enough time to recite the berachot of keriyat shema or say shemone esre (and in either case be finished by the end of the 4th hour), but they can't do both.
What should this person do?
After the 4th hour, they can't recite the berachot of keriyat shema at all (let's assume it's not ones, and be able to answer for sefardim as well as ashkenazim), but there is an opinion (the Rabbis - see Berachot 26a), as well as other provisions to say shemone esre up to chatzot (halachic midday), which one might be allowed to rely on in this case. Or one might argue that none of the above would apply in this case, and therefore the question becomes what is more important, berachot of keriyat shema, or shemone esre? Maybe the argument is that one is still before the 4th halachic hour is over, so one should endeavour to follow standard procedure.
I note that I am uncertain if, a few minutes before the 4th hour ends, one is allowed to start berachot of keriyat shema if one isn't going to be able to finish them by the time the 4th hour ends. That might be pertinent to answering this as well.
Thank you everyone for considering this question.
1 - Mishkanot Yacov O'C 80 states they can say the berachot even until chatzot, and Mishna Berura 71:4 and Beiur Halacha 58:6 says you can rely on this. This may complicate this question.
2 - Shulchan Aruch O.C. 58:6, Or Letzion 2:6:4, Yalkut Yosef 89:5
3 - Rif 18a, Rosh 4:1, and Rambam (Tefillah 3:1), Tur and Shulchan Aruch 89:1
