Does the fungus that cause athlete`s foot transmit easily through water eg shared bath/swimming pool?

Ideal environment. Fungus is a ground loving organism. It thrives in a warm moist dark environment, this is why antifungal powders can be so effective for controlling athletes foot infections. Control the moisture and usually you can control the fungal infections. Routine thorough cleaning of community areas is obviously also helpful.
Yes. It can be spread on tile surfaces. Wear sandals and clean the surface with disinfectant to prevent infection.