Hi, I think it's going to be really hard to say. One important thing is the situation before you initially fill the tanks - if they have been wet, and have possibly had some moderate amount of bio-organism growth going on, then it will start much sooner than if the tanks had been thoroughly dry for some period of time.
My experience comes from high volume photofinishing, where we would have problems during our busiest seasons. At those times, we'd sometimes have to run processing systems around the clock - 24 hours a day, and biogrowth would tend to get out of hand in the wash tanks. Other times of year, no problem - the tanks would be drained daily and have time to thoroughly dry before refilling the next day. When biogrowth DID start, I should probably say could be seen by eye, our recourse was to drain tanks, refill and dose with Clorox bleach for perhaps 30 minutes, then drain again and refill for processing. In the industry there was research into biocides that were not detrimental to photoprocessing, but these were pretty expensive.
In the US, the most common water supply treatment is chlorination, but this is gradually lost on exposure to air. So once the water has been in open tanks for a while, the protection is lost. And my personal interpretation, with no solid evidence, is that biogrowth, perhaps from organisms in the air, start growing right away, but take some time to reach an objectionable level (growth is likely at a geometric rate).
I would suggest that at the end of a printing session to add a small amount of "Clorox" bleach to the open water tanks - it will hopefully hold the biogrowth at bay and be gone by the next day. But if there is any trace of fixer (thiosulfate) in the tank, it will neutralize the bleach. If you do a lot of printing, it's probably preferable to completely drain and allow the tanks to thoroughly dry daily. Sorry there's no easy answer.