The iPod Nano can be knocked around pretty good and still keep going, but getting them wet is another story. The units have moistures sensors (much like nearly all of Apple's handheld line) are not to be got wet. The phone strap would provide some protection against this, but it doesn't make it water resistant or turn the device into an actual wristwatch capable of getting weathered. Washing your hands with the phone attached to your wrist would certainly get water in the unit (potentially leading to damage and inoperability) or at the very least, trigger the moisture sensors.
An iPod Nano is still an electronic device that should be kept clear of all water and as much perspiration as possible. If the moisture sensor is triggered, your warranty is void. With that said, the units are sealed quite well and would typically only allow water in the headphone jack and the proprietary cable connection port at the bottom (these are were the moistures sensors are located).