These languages don't belong to Indo-Iranian group like every other language in Europe.
I like to speculate the language came to northern Europe with Huns who, according to Procopius (I believe it was him) controlled Scandinavia too and there are many sagas of Norsemen fighting Huns there, such as Hlöðskviða.
The modern Finns show Nordic genetic, yet the language is Uralic and perhaps the first Finns were only the ruling class as it was often the case with Huns.
Obviously, there are other more widely accepted theories.
???
Blood flows like a river.
The Huns didn’t had a single language depending on the timeframe their tribes would’ve used Turkic, Romance and Germanic languages, while they possibly had some Uralic tribes in the mix too the timeframe doesn’t makes sense.
Ugric/Uralic languages could have spread form the Eurasian Arctic and Sub-Arctic regions and while Finnish and Hungarian are Ugric/Uralic they aren’t actually directly related which might mean a different starting point.
Hungarian for example is closer to Khanty than to Finnish suggesting a path of migration that leads them into modern day Russia and then back into Eastern Europe.
The exact origins are unknown there is strong evidence for both European and Siberian origin theories however the languages existed in Europe well before the Huns came so what ever the original homeland was the Huns are not the source of these languages in Europe or Russia.
Ptolemy already wrote of Khunnoi (possibly Huns) in second century.
Huns were heterogenous groups of many subjugated tribes of various races and ethnicities. What they had in common was that the ruling elite claimed tradition with Xiongnu or their political system resembled it.
What I thought Finns were Tacitus's Fenni, who were likely Germanic and have somewhat acquired the language from their conquerors who came from steppe. I'm not referring to Attilla's huns. There were nations coming from steppe for centuries prior.
I speculate, we may never know for sure of course.
The Huns or any other tribal group brining things through conquest doesn’t makes that much sense.
Finnic/Baltic-Finnic languages (Finnish and Estonian), Sami languages and (true) Ugric like Hungarian and Khanty are distinct groups that while might have had a common ancestor that would be predating the Huns even further (~7000 years ago) then you have Samoyedic languages which are also Uralic but tend to be looked at even more separately due to the differences despite Samoyedic meaning Sami people.
Samoyedic is the language used by the native Siberian and Russian Arctic tribes basically the Inuit/First Nations of Russia.
If anything I would bet on the fact that it’s more likely that the Uralic languages spread from the Eurasian Arctic south into Siberia and the Ural during say the last ice age than they were brought into Europe by the Huns circa -+couple of centuries 0AD.
The Uralic languages (namely the Finnic and Saamic branches of the family) were already spoken in what is now Finland, northwestern Russia and Estonia in the centuries BC. The Uralic languages had spread gradually over the area both through migration of a hunter-gathering population from Central Russia and language shift to Uralic from substrate populations around the eastern Baltic Sesa.
The Huns, on the other hand, were a federation of steppe pastoralists pushed into migrating west by climate and demographic factors. They spoke different languages entirely, and as steppe pastoralists, they were not very interested in the areas where Finnic and Saamic were spoken. They were hardly even interested in areas of Central Russia just north of their path of migration.
And most other languages in Europe belong to the Indo-European family, not the "Indo-Iranian group" (though there is a Central Asiatic and Indian-subcontinent top-level branch of the Indo-European family known as Indo-Iranian).
Eh, not quite: The Indo-Iranian group is one sub-branch of the Indo-European language family, and very few European languages are in that sub-branch. Further, there is at least one other non-Indo-European language in Europe: Basque, which is a linguistic isolate, with no known relationships to any other language, living or dead.