風邪は万病の元って英語でなんて言うの？

In order to explain that a cold can lead to many kinds of sicknesses and diseases, you can state this very easily by saying "A cold can lead to many kinds of sicknesses/diseases." or "A cold can lead to many kinds of diseases". Both these sentences directly state the point that you are trying to communicate, which is that a catching a cold can lead to other issues.

These sentences can be used to describe what effects a cold can have. Or they can be used to describe how some diseases/illness occur.

When saying "A cold can lead to any illnesses'' you are expressing that a cold can lead to sicknesses and diseases. When one says "illnesses'' sickness and diseases are implied because the term "illness'' refers to a sickness, disease, ailment, disorder, etc.

Colds are quite common, and often people don't think that they could lead to something more serious, especially if they are not taken care of properly. Colds often start in a simply way; a stuffy nose, a cough or flem, that is why we often say "a simple cold". Simple ailments can lead to other sicknesses if they are not cured well. Therefore, it is possible that simple simple colds may be the cause of other diseases, if they manifest badly. Some very serious diseases start out with a simple cold and many people may think that it is just the common cold, when it could be something else, more serious, like a sickness, a disease, an illness, a virus or any other serious condition.