We know the extent of the Laurentide ice sheet at the last glacial maximum. it covered North America down to the great lakes.In Europe it covered Scandinavia and the West of Russia as far as Moscow, It did not cover Siberia. The polar ice sheets are centred on the poles because of the reduced insolation. Weather patterns that might affect the extent of ice cover are also driven by this. And in any event there is no possibility of ocean currents comparable to the gulf stream that could have kept Siberia free of ice.
The centre of the Laurentide ice sheet was around where Hudson Bay now is. The inescapable conclusion is that the North Pole must have been there, not in its current location. The physics of the earth in the solar system dictates that the earth as a whole cannot change it’s rotational axis except through gravitational forces that would need planetary interactions of the kind that may have happened during the formation of the solar system. Therefore the only way that the pole of the earth’s rotational axis could have been in the Hudson Bay is if the crust of the earth is able to slide over the interior of the earth.
A much lesser force might be available to cause this to happen, if the weight of the Laurentide ice sheet was not equally distributed over the pole. This was the case. Polar ice can achieve a thickness of thousands of feet on land, but over the sea, unless there is an ice shelf, it floats, and hence has no greater weight than the ocean. The part of the ice sheet over Europe was considerably smaller than that over North America, providing an imbalance that would apply a force to the crust as the earth rotated. It would tend to drive the ice sheet towards the equator in the direction of where it was heaviest. In other words, North America would have been pushed South, pulling the Arctic ocean over the pole. This was presumably a slow process and as it happened from 20KBP until probably 11800BP, the ice sheet would have melted, being no longer at the pole, reducing the force, allowing the crust to settle where the pole now is.
This may be fortuitous – see the insight about our current climate.