For mechanical reasons, my wet-day bikes are both unavailable, so today I rode to work on my nice road bike. It has an "ass-saver" but nothing more. The rain was torrential.
I got wet. I got so wet my shoes had puddles inside them. My work pants were so dirty with road grime that they looked like I'd had a fall.
My gloves ran water like a tap if I made a fist with one hand. My under-thighs caught a lot of spray water from the rear wheel resulting in potential chafing.
There was road dirt under the saddle, and all round the seat post. The rim brake tracks looked like a desert, and worked like sandpaper any time I braked, so there was more coasting to a stop than normal.
Once at work, I had a hot shower and changed into spare clothes and shoes I keep at work. I also wrung out my clothes in the shower, removing the worst of the water.
Over the course of 8 hours, most of my clothes dried well enough. The gloves were the wettest part at noon, so I put them in a warm for the afternoon and they still didn't dry fully. Shoes also were not perfectly dry but fresh socks helped.
At home that night, I put all the wet clothes into the wash, the gloves and shoes needed for tomorrow go in a warm spot to dry.
One day of wet riding is not a big deal as long as you're prepared.
A week of it, or several months of winter is harder to deal with.
In lieu of mudguards, you can wear overtrousers and a waterproof parka. However I find that they don't breathe very well and I get "parboiled" in sweat.
Good overshoes help a lot - the neoprene ones are fine in the cold but useless in the wet. Rubber overshoes, or ride in gumboots works better in the rain.
Gloves are a mixed bag - thin enough to feel means not waterproof or warm. Thick enough to be warm, and waterproof both mean poor sensitivity. Both warm and waterproof feels like wearing a chicken on each hand.
Additionally, mudguards/fenders keep the road grime off your chain and BB by fully wrapping the rear wheel's top half. If you don't have guards and ride a lot of roads in the wet, you'll end up doing chain/transmission/derailleur servicing more frequently. 3/8 coverage front and 1/2 coverage rear guards help a lot.