Athlete’s foot is caused by a fungus that thrives in damp conditions, like the bathroom.
That having been said, it’s very rare, although not impossible, that a young child would catch athletes foot, from the bath or from anywhere else.
If you are concerned, make sure that you wash the bath thoroughly before bathing your child in it. Make sure that you dry your child’s feet carefully after the bath, especially between the toes, where the fungus most often takes hold. Try to let your child’s feet air dry properly before putting closed shoes on, and your child should be fine.
Athlete’s foot is an itchy, flaky skin condition that occurs on the feet, and it is caused by a fungus. It’s usually harmless, although it can be irritating, but there are commercial preparations that will sort out an infection within a few days to a week.