Finnish and Hungarian are Uralic languages. The evidence is pretty plain that this family did originate in the Ural Mountains region just as the name says.
Sami (formerly called Lapp) is another Uralic language, related to Finnish, Estonian, Hungarian. The Sami people themselves may be the descendants of you Neolithic Reindeer Hunter culture. (What’s that about hunting reindeer? I thought they herded them.) Whether or not that’s so, their language is not indigenous to Scandinavia. I’ve heard a theory that they were a different race who took to speaking a Uralic language after contact with Uralic peoples, and lost their original language.
This kind of language transfer is not unknown. The Orang Asli, negrito aborigines of Malaysia, now all speak Austroasiatic languages. The Negrito being racially and culturally different from the main Austronesian speakers (like the Khmer & Viet), they have lost their original language after contact with Austroasiatic speakers. The Negritos in the Philippines speak Austronesian languages like the Filipinos around them. One idea of what the real original Negrito language might have been like are the still-living languages of the Andaman Islands, populated by Negritos who were isolated from the rest of humanity for many thousands of years.