Does Earth's magnetic field have any effect at all on the climate?
It keeps our water and atmosphere in place.
Off hand, I cannot think of any magnetic component relating to our weather much less our climate that would care if the magnetic north is in Canada or Siberia. And, were we tracking this before 1900 to know the previous pattern? There is no reason to think it couldn't turn around tomorrow and start heading back south again and no reason to it hasn't always been moving around like this or think that it has always been heading in a straight line and will continue into Siberia. If we did, we would have to accept that the mag north was in the United States back during the Dark Ages!
The Earth's core is a semi-molten rotating body that acts as a dynamo generating a magnetic field and is constantly wobbling at least magnetically. Of greater concern is that it doesn't reverse polarity or slow and stop. While it doesn't keep the water and air directly in place, it's total absence would mean unshielded exposure to the deadly solar radiation which would sterilize the surface of the Earth of life in a rather short time. Much later over millions and billions of years the streaming particles might strip away all of the atmosphere and some of the ocean.