The new latitude should be easy if we can determine the polar star and measure its angle. If the ham radio system is up and can broadcast a time hack at its local solar noon, the rest of us could then back into our own relative longitude by measuring that portion of a days rotation to our local solar noon. Obviously we'll also need to check to see if we still have 24 'hours' in a day after we resume rotation. I think we have to assume everything will be perturbed.
Probably wouldn't hurt to make a startup scenario for redefining our world after the pole switch. How could we get a digitized map of the new world assembled? The ham radio net could be very helpful in determining data points and centralizing them. A good supply of cheap plastic sextants would help in determining local solar noon and maybe the pole star elevation.
We may also abruptly switch seasons. Tracking the sun's path will allow determination of the new equinoxes and solstices. Critical for planting and growing.
Offered by Jack.