The ocean floor between Australia and New Zealand will shift some, and sailors will have to relearn the currents. Australia will suffer damage, but it will look basically the way it does today. It will also have a new location.
Since this continent won’t break up, will there be safe places for the people to go?
To stay away from the coasts and to stay out of the desert. Considering that most of the major cities are on the coast they will suffer much damage from tidal waves. The central part of the country, which is all desert, will experience fierce storms and flash floods. There will be climatic changes all over the world.