They have several genetic adaptations to life in the far North (See Scandinavian porphyria, dwarfism, resistance to cholera, black plague, etc.) that would probably take longer than a mere 5 or 6 thousand years to develop and spread.
Authorities cite anywhere from 7 to 9 different full-blown Saami languages, all vaguely related to Finnish and other Uralic/Altaic languages. No doubt Turkish/Mongol words have infiltrated the Saami languages, as have modern English words, but the grammar is different.
If anyone wanted to make that leap into determining what language was used 20,000 years ago, he would be well advised to study Saami since it may be based on linguistic traditions 35,000 years old.
The article mentions Finnish being in the same category as Hungarian, which I knew. I also have heard that these two are related to, of all things, Korean.