The basic answer is "I didn't have a shower this morning" (I think "take a shower" is only US English... correct me if I'm mistaken). It's a simple statement about a past event. I'll try to give a context for the other sentence, as per Randy's comment: A: "Were you having a shower this morning? I heard water running." (This also means the running water was heard for some time, of course.) B: "I wasn't having a shower this morning. I was just washing the dog." (You see this is a response to a specific question.)

The sentence, "I didn't have a shower this morning would be more correct, but the most correct way to say it would be to say,"I didn't take a shower this morning"

More normal is "I didn't have a shower this morning", but you could also say "I wasn't having a shower this morning" in some context.