Getting wet is only a problem if it is also cold and/or windy.
Getting cold when it is wet is frequently a problem - I cycled up the west coast of the South Island of New Zealand in heavy rain - the air temperture when I was sheltering at bus stops was positively tropical - however I did not need to cycle very far without waterproofs to be decidedly on the wrong side of cold.
I always carry a proper jacket, leggings, a hat (normally a baseball one that I have treated with silicone), some decent gloves(s) and overshoes - plus on my last long tour I had a pair of shorts that I had made from a damaged set of legging - useful when the rain is coming and going