Other countries mange this with proper clothing and a variety of alternative public transit options.

Rain Capes are a popular solution for rainy weather when cycling.

Or you chose to avoid the bike that day and take the bus/streetcar/metro/etc.

I live in NYC and by far my favorite aspect is being able to decide between a variety for transit options that best suit the specific trip I’m making. For example, I typically commute by bike, but if it’s raining I can easily switch that trip to be on the subway.