No. This is an old wives tale. Fill up the bath and the basins in your house and see whether the water always runs out the same way in all of them. Then ask your Australian friends to do the same. The Coriolis effect of the earth's rotation is negligible compared with other things such as the level of the bath, irregularities in the plumbing and - most importantly - the vortices you cause when pulling the plug out.
Some years ago universities in London and Australia carried out experiments using baths in which all irregularities were engineered out, including the disturbance caused by pulling the plug. The experiments were abandoned when they discovered that it took eighteen hours for the vortices caused by filling the bath to disperse! A friend of mine recently told me that he had noticed the difference when at the Equator and had crossed from one hemisphere to the other. I had to point out that the Coriolis effect, if it existed in baths, would be at a maximum at the poles and zero at the Equator!