The Rambam, in Hilchot Nachalot 6:9, rules that:
הגוי יורש את אביו, מן התורה; אבל שאר ירושותיהן, מניחין אותן לפי מנהגם
A non-Jew inherits his father according to Torah law; however, the for the rest of their inheritances, we leave them according to their customs.
The ruling regarding a father to son is based on the Talmud Kiddushin 18a which derives this rule from the verse in Devarim 2:9:
כִּי לִבְנֵי-לוֹט, נָתַתִּי אֶת-עָר יְרֻשָּׁה
because I have given Ar unto the children of Lot for an inheritance.
The Maggid Mishnah on the Rambam points out that the reason we leave the law up to the local custom in the other cases is:
שלא מצינו שיהיה סדר נחלות אלא לישראל בלבד
for we have not found an order of inheritance except for Jews
implying, that the difference is as regards order of inheritance, not the existence of the right of inheritance. As such, it seems that the son, or whoever else happens to be the one inheriting, acquires the property immediately parallel to the rules of inheritance by Jews.
The Minchat Chinuch (Mitzva 400) questions this read and suggests that indeed with the exception of a father-to-son, all other non-Jewish deaths are like the deaths of a relative-less convert whereby all their property is rendered ownerless.
Rav Shlomo Zalman Aurbach (Minchat Shlomo 1:86) rejects this Minchat Chinuch, affirming that the Rambam held that the non-Jewish inheritance customs do affect Jewish civil law and that the other inheritances work as per local custom.
He also notes that even according to the Minchat Chinuch, the rule Dina DeMalchuta Dina would ensure that it remains forbidden to take the property of a son-less non-Jew.
Some (Peri Yitzchak 2:60, Har Tzevi Yoreh Deah 223) debate if the above discussion about fathers and sons applies equally to mothers and daughters.
I finally note that the Shulchan Aruch excludes the part about local custom in his ruling on the matter in Choshen Mishpat 283:1.