Fortunately, you don't need showers. Bicycling != sweaty. And sweaty !=stinky.
People stink because they cover themselves in chemical perfumes (perfumes, soaps, shampoos, moisturizers, etc) that have limited 'life' before they start to decompose. People stink because they use fabric softener on their clothing (see above...also, fabric softeners are basically fat. Which goes rancid...) People stink because they toss their sweaty clothes into a hamper instead of airing them out.
I'm American. I ride a bike every day, in a city that goes from 0 degrees to 100+ degrees. I don't understand this obsession with sweat. If I'm going somewhere and can't show up sweaty, I slow down, or I get there a few minutes early and cool off/dry off just by...uh...standing around...
People seem to forget that biking is more efficient and thus for the same energy used walking, you can go several times faster, which means for the same energy you are less likely to get sweaty.