Question:
If there is no time to wash hands or say birkas hatorah and still squeeze in krias shema before the zman ends, may one say shema before taking care of the other 2 activities?
Answer:
If it is so late that you can’t even say birkas hatorah, you may say kriyas shema first. This is because you are now saying kriyas shema in order to do the mitzvah, without the mitzvah of talmud torah.
If there isn’t time to wash your hands, then regarding saying kriyas shema which must be done now, it is as if you don’t have water. Therefore you should wipe your hands on something, (preferably something disposable, as we should try not to touch things before washing our hands) and say kriyas shema.
This answer should not be taken that this is what a person should do l’chatcila, and this should not be taken as a viable option to wait for the last minute, (or to be lazy and not get out of bed to wash your hands). However if it already happened, this is what should be done.
Sources:
M:B 47-17, Yosef Ometz 66, Eretz Tzvi 1-19, Btzel Hachchma 1-1, O:CH 4-22.
