The net change in the position of the Earth's rotational axis is about 37 feet. The largest annual change is about 7 inches.
If you move over 37 feet, the climate doesn't change. It changes even less per year if you only move 7 inches.
The research paper itself is here:
While the Earth's magnetic field is weakening a bit and its magnetic axis is shifting somewhat, magnetic field polarity changes have no effects on climate on the timescale of human lifetimes because air isn’t ferrous. The effects on hand-held compasses are insignificant. For purposes of electronic navigation, changes in the position of the magnetic poles are constantly updated in navigational databases.
"The last time that Earth's poles flipped in a major reversal was about 780,000 years ago, in what scientists call the Brunhes-Matuyama reversal. The fossil record shows no drastic changes in plant or animal life. Deep ocean sediment cores from this period also indicate no changes in glacial activity, based on the amount of oxygen isotopes in the cores. This is also proof that a polarity reversal would not affect the rotation axis of Earth, as the planet's rotation axis tilt has a significant effect on climate and glaciation and any change would be evident in the glacial record."
"The science shows that magnetic pole reversal is – in terms of geologic time scales – a common occurrence that happens gradually over millennia. While the conditions that cause polarity reversals are not entirely predictable – the north pole's movement could subtly change direction, for instance – there is nothing in the millions of years of geologic record to suggest that any of the 2012 doomsday scenarios connected to a pole reversal should be taken seriously."
"What would happen if the magnetic field of the Earth suddenly changed?