If Klal Yisrael would all die, ch"v, then would Edom become obligated in Tora uMitzvot because of Yibbum? (because Jacob's/Israel's brother is Esav, Edom.) Or would chalitzah happen because no other nation is able to serve God like we are (cf. midrash about the Edomites and Ishmaelites rejecting the tora (and thereby rejecting God) because stealing and sexual immorality are forbidden)? When the Torah was given, and the Children of Israel were under the mountain (tachat hahar), Aruch Hashulchan and others compare Har Sinai being above us to being like a chuppah, and that kabbalat hatorah is like marriage. Obviously, generations of the Children of Israel have died out, but they had offspring. But if there is no offspring, then what? (Tangentially, it is a bit interesting because although Hashem is our Parent, He is also our Spouse. A bit incestuous if you ask me. Also, Edomite theology (if you say it equates Christianity) already says that God rejected Israel.)
כִּי אֵל רַחוּם יְהוָה אֱלֹהֶיךָ, לֹא יַרְפְּךָ וְלֹא יַשְׁחִיתֶךָ; וְלֹא יִשְׁכַּח אֶת-בְּרִית אֲבֹתֶיךָ, אֲשֶׁר נִשְׁבַּע לָהֶם. (דברים ד:לא)
For the LORD thy God is a merciful God; He will not fail thee, neither destroy thee, nor forget the covenant of thy fathers which He swore unto them. (Deuteronomy 4:31)
A gentile brother of a Jew (say the Jew is a convert) wouldn't be obligated in yibbum because he was never obligated in Torah. Even if he somehow were, the Jew is no longer his brother, halachically speaking, so yibbum wouldn't apply. (Per @DoubleAA, see Rambam: Issurei Biah 14:11-12 and in English.) And yibbum is a mitzvah that applies to individuals, not whole nations.
So no, Edom isn't an individual, and even if that somehow doesn't matter then Edom is no longer Israel's halachic brother, and even if that somehow doesn't matter then non-Jews were never commanded in yibbum.
God is often called our husband; I don't recall ever seeing him called our wife. If we were to die, then, presumably there would be no yibum.