Given the paucity of information about this part of the world during Late Antiquity and the Early Middle Ages (indeed, we cannot even say with certainty what sort of language groups such as the Huns and the Avars spoke), it may be difficult to discuss this topic, but we can certainly try!
If, for whatever reason or reasons, the early Slavs were not able to capitalize on the power vacuum left by the invasions of the Huns and migrate across much of Central Europe and Eastern Europe, assimilating the inhabitants... well, who would have most likely taken their place over the long-term as the dominant ethnolinguistic group in Eastern Europe? (Obviously not
every Eastern European country is majority Slavic, but I think it is fair to describe Slavs as dominant in the area.)
Here are some thoughts...
Baltic peoples: See the explanation given by
@Augenis below.
Finno-Ugric peoples: The Magyars obviously made a huge impact with their own migration into Europe, but there was also a significant Finno-Ugric presence in the Baltic and northern Russia at this time.
Germanic peoples: East Germanic peoples such as the Goths were numerous in this area before the Huns pushed many of them west, and centuries later kingdoms of West Germanic peoples (for instance, the Franks) and the Norse (Varangians) would attempt to push into now-Slavic lands in Central Europe and Eastern Europe.
Hellenic culture: The Eastern Roman Empire remained a significant entity for long after the fall of the Western Roman Empire, so in the absence of powerful Slavic kingdoms, could they have expanded further into Eastern Europe and assimilated many of the inhabitants?
Iranian peoples: There were many Sarmatians in Eastern Europe until they were assimilated by Early Slavs.
Romance speakers: Though the exact ethnogenesis of the Romanian people is a matter of debate, it is clear that there were speakers of Vulgar Latin in this area at the time.
Turkic peoples: Many Turkic peoples migrated into Eastern Europe throughout the Middle Ages. In the early part of this era, the Bulgars and Khazars were significant powers in the area.
Some combination of the above: In this scenario, no one group of people becomes as dominant in Eastern Europe as the Slavs would in our world.
Other: Some ethnolinguistic group that I did not mention.