Parking yourself nude under a cold shower will keep you alive, assuming you don't run out of water.
There's no need for the bath. The shower will probably work better actually since it's constantly disposed down the drain after picking up some heat, and replenished with subterranean-temp fresh water running across your person.
I'm not sure a bucket would be enough to cool a person down if they have no other form of heat dissipation available to them - specifically I'm not sure feet alone could dissipate heat fast enough. If the water is too cold (around 70°F IIRC), your body would constrict the blood vessels to your hands and feet, limiting how much heat could be moved to those extremities from your core.
Hence mentioning a bath tub.
Some interesting reading to go along with this topic:
You’re going to need to set some practical constraints here.
Can you survive wet bulb temperatures given a bucket of water indefinitely? No. Can you survive working hard labor with the water bucket? No. Can you survive wet bulb events significantly higher than the threshold? No.
Can you survive several hours in a typical (current typical) wet bulb event? Yes, definitely. Bucket of water is fine.
> Can you survive several hours in a typical wet bulb event?
I'm going to vote maybe. At best, it'll be close.
Resting metabolic output is about half the active value I provided (per NASA's PDF). That means for a bucket of water that starts at 32°, you have 5 hours before the bucket is also 95° if you discount ambient heating from the air, circulatory constriction, and assuming no prior conditions.
Since you can't discount those (and are unlikely to find barely-not-frozen water for an entire city's population), survival for more than an hour or two is nowhere near guaranteed.
> survival for more than an hour or two is nowhere near guaranteed.
Does that pass your personal sniff test for reasonability? Have you ever soaked in a 104°F (40°C) hot tub for an hour or more? Was your survival in serious doubt?
Wet bulb temperatures in excess of 35°C need to persist for around 6 hours to represent a serious hyperthermia risk. This is not a "you will die inside of 1-2 hours" scenario.
There's no need for the bath. The shower will probably work better actually since it's constantly disposed down the drain after picking up some heat, and replenished with subterranean-temp fresh water running across your person.