If one woke up in the morning with literally moments left prior to the end time of Shema, may one recite Kerias Shema prior to washing his hands?

Yes. In such a case one is to quickly wipe his hands on any item within reach, including one’s blanket, and then quickly recite the Shema. One is to then wash his hands. If you are unsure as to how much time is remaining and you don’t want to take a chance to wash, then you may say the Shema on condition, and then wash and then say it again. [Vetzaruch Iyun whether in such a case one should say all three paragraphs of Shema, or only the first two paragraphs. Seemingly, however, one should say the first two, and not the third.]

The reason: The prohibition against saying Pesukim prior to washing is based on the Zohar, while according to the Talmud it is permitted, so long as one slept with clothing or wiped his hands beforehand on an item. Now, certainly, one should not nullify a Biblical command in the Torah to say the Shema on time due to a Zoharic prohibition. Furthermore, the Alter Rebbe himself rules that when there is no water available, one should not avoid learning Torah prior, and hence certainly one must say Shema beforehand if the time is passing.

Sources: See regarding saying Pesukim before washing if no water is available, and certainly the same applies here for the sake of the Biblical Mitzvah of Shema: Admur Basra 1:7; Kama 4:3; 47:9; Siddur Hilchos Netilas Yadayim See regarding the general prohibition of saying Pesukim before washing: Basra 1:7; Kama 1:7; Siddur; Sefer Haminhagim p. 4 [English]; Mahram di Lunzano in Derech Chaim; Or Hachama [Avraham Azulai]; Birkeiy Yosef 4:8; Zechor Liavraham 4:50; Yifei Laleiv 47:9; Kaf Hachaim 4:107; M”B 4:61; Rama 47:13 in his Lechatchila opinion. See regarding the general prohibition of touching clothing before washing: Basra 1:7; Kama 1:7; Siddur; Seder Hayom; Peri Eitz Chaim; M”A 4:1; Beir Heiytiv 4:1; Ben Ish Chaiy Toldos 6; Olas Tamid p. 17a; Tzvaas Rebbe Eliezer Hagadol; Sheivet Mussar 16; Sefer Haminhagim p. 4 [English];  See regarding the three paragraphs of Shema: Admur 46:1; 58:1; 60:5; 63:6; 67:1; 128:42