Cold medicine itself does not affect menstruation, but having a cold can affect your constitution, which can influence your menstrual cycle. Many factors can impact menstruation, such as physical condition, excessive staying up late, fatigue, high stress, or extreme dieting or gaining weight. If your period is delayed, the first thing to consider is whether you might be pregnant, and then consider other factors that might cause the delay. If it is delayed by a week, you must first perform a pregnancy test.