There are a couple of issues that are the primary concern in this situation.
The most common cause of a productive cough upon awakening that also may cause a lesser amount of cough throughout the day would be allergies or acid reflux.
Allergies can have many different patterns of symptoms, but certain substances to which someone can be allergic are more of an issue when someone is lying in bed, such as dust mites. So one of the common patterns would be a worsening of symptoms while asleep, so would be most noticeable when you wake up.
Acid reflux also frequently gets worse at night, because gravity is not helping to keep stomach contents where they belong when you lie down. I asked about the lansoprazole to see if you were already diagnosed with acid reflux. While the most common symptoms associated with acid reflux is heartburn, it is also fairly common for acid reflux to present solely with lung symptoms, including cough, shortness of breath, or wheezing.
I ask about the fever, because the associated night time sweating can be concerning about some less common conditions. Certain chronic infections of the lung can present with night sweats and a morning cough. If there is a fever associated with the night sweats, that would be more concerning, but since you have not had fever, that is a relatively good sign.
While an evaluation would be the est method to clarify the cause of symptoms, there are a couple of home interventions that can be tried to see if the symptoms can be eased. Taking an antihistamine at bedtime can help with night time allergies, but it would be better to use a longer acting antihistamine. The Benylin is a fine antihistamine, but if taken at bedtime, it usually will not last through the night. A longer acting antihistamine, such as loratidine or cetirizine, may work better. For acid reflux, since you are already taking lansoprazole, another home intervention that may help would be placing 4-6 inch blocks under the feet at the head of the bed for elevation, which can help to restore some benefit of gravity. Note that sleeping on extra pillows is not as effective, as it causes the body to bend int he middle, which can increase pressure int he abdomen and exacerbate acid reflux.
If one of these home interventions help the symptoms, then that would clarify the cause of the symptoms. If these interventions are not helpful, then it would be appropriate to be seen by your doctor for a proper evaluation.
If I can provide any additional information, please let me know.