Following the general naming convention of still-practiced forms of polytheism and animism, the Norse religion would probably be called "ásvegr". Just following the norms of polytheistic and animistic religions in naming themselves. They almost always call themselves the "way of the Gods" or "the way of the spirits". That's what ásvegr means. This word is actually used in Skaldic poetry and the sagas to refer to Norse Polytheism. So it is almost certainly the correct name for those beliefs.
The modern term ásatru was created from Old Norse roots, but was not likely used at that time. And the modern Asatru belief system is a neopagan movement religion, which doesn't take into account the influence of Siberian and Uralic religions on the Norse beliefs.
If one did take North Eurasian influences into account, the faith would look drastically different. Especially since the Mesolithic Urheimat of the Uralic peoples puts them into direct contact with peoples like the Ynesians, Chukchi, and Kamchatkan groups, and this shows up in the Eddaic literature from Iceland and the Circumpolar Bear Cult practices such as the Feast of Otso described in the Kalevala and the Iyomante ceremony from Sakhalin and Hokkaido, as well as the Na-Dene version of the Spirit-Bear mythology...
Uralic, Turkic, Mongolic, Japonic, Dene-Ynesian, and East Siberian forms of Shamanism are all very clearly similar, and their Urheimati are all within an easy distance of one-another. So while the Ancient and Classical Norse people and their culture is broadly Indo-European, their religion has a very strong indication of Siberian influence, and this is not only by comparative study of their mythology, but also evidenced in their language, their concept of destiny, their systematic code of honor, and many symbolic motifs.
