There is a three way argument what was the impurity of those who asked for a Pesach Sheni.
Rabbi Yossi HaGalili said that they carried Josef's bones
Rabbi Akiva said that they removed Nadav and Avihu's corpse
Rabbi Yitzchak said that both of those could have purified themselves in time, as the Jews camped for close to a year by Mt. Sinai, rather they had an emergency Meis Mitzvah.
What would Rabbi Yossi HaGalili and Rabbi Akiva answer?
Rav Hirsch points out that in Pesachim 90b it says that they had already undergone the Parah Adumah sprinkling (for the seventh day) and they were only waiting for sunset. Had they been allowed to join a group (that would be sacrificing) they could have eaten. however, that was not allowed.
One answer for Rabbi Akiva is that Nadav and Avihu died on the eighth of Nisan rather than the first (Rosh Chodes) as there is a machlokes as to when exactly the mishkan was first set up. One statement is that they started setting it up on Rosh Chodesh Nisan and יום השמיני was on the eighth day of Nisan. This meant that they would not have become tahor until after sunset and it would have been too late to be part of the korbon Pesach.
Others would say that the deaths were on the first of Nisan and one possibility is that the Parah Adumah was not yet ready and that caused the purification to be delayed.
A possibility for Rabbi Yosi Haglili is that they had to continue attending the coffin of Yosef while the mikdash was being set up and taken down (similar to the first answer) or that they might not have finished arranging for the Parah Adumah (as with the second possibility).
In either case, we see that there is an answer for both Rabbi Akiva and Rabbi Yosi Haglili.
Although some commentators (most notably Malbim) suggest that Rabbi Akiva necessarily holds that Nadav and Avihu died on 8 Nisan, there is no indication that he disagrees with the generally accepted view in Chazal that their deaths occurred on 1 Nisan. As to why Mishael and Elzaphan would still be Tamei Meis on 14 Nisan, a possible answer is that they buried Nadav and Avihu far away from the vicinity of Har Sinai and it took them several days (more than a week) to return to the camp. This answer is given by one of the commentators on Masseches Sukkah found in Otzar Mefarshei HaTalmud page 831, but unfortunately I don't recall his name and don't have the sefer before me. Unlike others (Miriam, Aharon and Moshe), the Torah does not tell us exactly where Nadav and Avihu were buried; it simply says that Mishael and Elzaphan took them "Michutz LaMachane." That indefinite phrase could even mean Har HaZeisim outside of Yerushalayim, the place where Tzadikkim throughout the generations were buried--Yerushalyim corresponded to Machane Yisrael in the Midbar, and thus fits the term "Michutz LaMachne." Moreover, Chazal call Har HaZeisim "Har HaMishcha," and Nadav and Avihu are called HaKohanim HaMeshuchim in B'Midbar 3:3. (The next verse refers to their deaths.)
There is no necessary contradiction between the views of R' Yossi HaGlili and R' Akiva--Mishael and Elzaphan could have been the men who dealt with the bones of Yosef--Mizvah Goreres Mitzvah. (In fact, they could also have been the men who found the Meis Mitzvah.) R' Yossi focuses on the fact that the original tumah from which they were never purified involved the bones of Yosef, while R' Akiva focuses on the cause of their inability to be tahor on 14 Nisan--the burials of Nadav and Avihu.
