Somehow I’ve managed to commute to work on a bike year round for 10 years now, and I’m in a customer-facing role.
When it rains I wear rain clothes (jacket, rain pants and over-shoes). I also have full fenders and a chainguard on my bike.
If necessary, I ride in cycling clothes, carry my work clothes and some deodorant in a pannier, and quickly change in the bathroom when I arrive. It takes 5 minutes.