The common cold is caused by various viruses. If there is a weather change that makes one susceptible to a weaker immune response or immune system and they catch a virus that causes a cold - then I can see you catching a cold as a result of the rain. Most water systems like your shower are purified and chlorinated.
Keeping this in view, can being in the cold with wet hair make you sick?
A wet head can make you sick. You'll probably feel chilly if you skip the blow-dry on a cold day, but not much else will happen. Again, colds are caused by a virus. Unless you are so cold that you get hypothermia, which could make you susceptible to infection, wet hair or clothes won't increase your vulnerability.
Is it bad to go outside in the cold with wet hair?
Of course, you can catch a cold or flu virus while outside, and cold and flu season does overlap with cold weather. Recent research suggests rhinoviruses, which cause the common cold, may thrive in low temperatures. But you won't get an infection simply because your hair is wet.