With a steady supply of cold'ish water I would guess it would do a decent job of keeping you chilled, and not on fire. Maybe? This is similar in function to systems they deploy on some firetrucks that could be in danger of getting trapped in a burnover. Of course, those keep a truck exterior cool, not an early 20th century raincoat.
Either way, it looks wildly impractical, and the user must have a hard time seeing much.
We're in a bushfire zone, even had one get close, late last year. We looked into suppression systems, and there's two types - one has a series of garden sprinklers positioned on your roof gutters or overhangs. They get turned on to create a large, extended fan-type spray of water. The idea is not to extinguish a fire, but to absorb the heat so it doesn't get hot enough locally to ignite your house.
The other type - which we chose - puts agricultural sprinkler heads on your roof peaks. Fed by a substantial pump from storage tanks ( 2 x 22,500 litre/5000 gallons), they throw in intersecting circles out to a distance of about 15 metres/50 feet. The idea is to saturate your roof and walls, and surrounding foliage sufficiently that it won't ignite.
My dad was a firefighter and showed me the various nozzles they used on the firehose. One of them worked kind of like this, except it sprayed out of the hose (obviously), and created a shield-like spray in front. It was used mostly for cooling down hot air and combustible gases. It also carried the benefit of soaking the corridors you were walking through. If you're caught in a blaze, nothing will help you, but for anything else, this would be pretty useful.
In an intense fire this just made thing worse. Especially for the person wearing it. Plus supplying water to this was not ideal either. Especially as the water used here is water that couldn't be pumped into the main hose.
This would make it so much worse. Water would just increase the heat transfer to the wearer.
Good home example, if you wet an oven mit and try picking up a hot pan with it, it will burn your fucking hand. If it was dry, it wouldn't transfer the heat as fast.