Problem identified! The consequences of a circa 1984 wiring job which would not meet current building codes, combined with sheer random bad luck.
Power runs out from the circuit breaker to the outside outlet which got wet. From there it goes up into a junction box in the attic and branches off to both bathrooms. With the outlet dead, no power makes it to either bathroom. Above the outlet is a light, which is on a different circuit from the outlet. There is a switch which controls power to both the light and the outlet. So with the switch on, the light still works even though the outlet doesn't, which is confusing as fuck until we determined that the light and the outlet are on different circuits.
Short answer: replace the outside outlet with a new GFI outlet and everything should be wonderful.
Cost to replace an outlet: ~ $10.
My contractor's paint sprayer hose coming loose in a freak accident and spraying paint everywhere, leading to the blown outlet: priceless.