I have always found the most success with a dollar-store bristle handbrush. As others have said, a bleach solution is going to be a good starting point. Turn on the vent and open the window when pouring it into the railings and – I say this from experience – if you aren’t getting the desired results, do not add ammonia to speed things up.

But basically, yeah, you want an old towel around for incidental splashes, but aim the brush such that the gunk is going into the shower, not creating a new hassle. I find that two hours is more than sufficient, and with good ventilation and you not being in the washroom, it’s a pretty harmless chemical that gets the job done even at full consumer strength. Adding water within reason retains the efficacy while making the whole affair a fair sight cheaper. Basically, it’ll smell like an indoor pool.