Just check the phylogeny. If a deep subclade, like Q1a2a1c, is found in Europe are the same as in Mongolia, then it means that the common ancestor for this subclade is too recent to be a Palaeolithic offshoot from Siberia to Europe. You have to look at the TMRCA otherwise it's impossible to determine the historical time frame for each subclade.
Obviously, if you found some Q*, Q1* or even Q1a* in Europe, then it could have come during the Ice Age.
It's case by case, as it depends on their subclade, but ultimately the patrilineal ancestor of haplogroup Q was Siberian and probably Mongoloid looking. That being said, it is not even sure that the Huns still looked Asian by the time they reached Central Europe, as they had been mixing with Europid/Caucasian people for many generations. What is certain is that the Vikings that brought Q1a to the British Isles did not look any more Asian than their I1, R1a and R1b comrades. Actually there is not much difference between Q, R1a and R1b in this regard.
The original R1 ancestors also had Mongoloid features. This is sure from the testing of the 24,000 year-old Siberian Mal'ta boy, who belonged to R1* and was of the Mongoloid type. The main difference is that R1a and R1b people migrated west earlier and therefore mixed with Caucasoid people well before people of haplogroup Q. But anyway it only takes a few generations for Mongoloid features to be completely faded by intermarriage.
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