Acute Bronchitis: This is the most likely diagnosis given the symptoms of a productive cough lasting more than a week and the presence of wheezes on physical exam. Acute bronchitis is a common condition characterized by inflammation of the bronchial tubes, often caused by a viral infection, which leads to cough and wheezing.

Asthma: Asthma is a chronic condition that can cause wheezing, coughing, and shortness of breath. It's possible that the patient's symptoms could be an exacerbation of underlying asthma, especially if the cough is productive and accompanied by wheezes.
Chronic Obstructive Pulmonary Disease (COPD): COPD is another condition that could present with a productive cough and wheezing, especially in smokers or those with a history of exposure to lung irritants. It's a consideration, especially if the patient has risk factors.
Pneumonia: Although pneumonia typically presents with more severe symptoms such as high fever, chills, and possibly chest pain, a mild case could present with a productive cough and wheezes, making it a consideration in the differential diagnosis.

Pulmonary Embolism: While less common, pulmonary embolism can present with sudden onset of cough and wheezing, among other symptoms like chest pain and shortness of breath. It's critical to consider this diagnosis due to its high mortality rate if untreated.
Tuberculosis (TB): TB can cause a chronic cough that may be productive, along with wheezing due to bronchiectasis or other complications. Given its potential for severe consequences if not treated, TB should be considered, especially in high-risk populations.
Lung Cancer: In rare cases, lung cancer can present with symptoms similar to those of a productive cough and wheezing, especially if the tumor obstructs an airway. This diagnosis is critical to not miss due to its significant impact on prognosis and treatment.

Cystic Fibrosis: This genetic disorder leads to the production of thick, sticky mucus that can cause respiratory symptoms, including a productive cough and wheezing. It's more commonly diagnosed in childhood but could be considered in adults with a suggestive history.
Bronchiectasis: A condition characterized by damage to the airways, leading to their widening and scarring, which can cause a chronic productive cough and wheezing. It's less common but should be considered in patients with recurrent infections or specific risk factors.
Allergic Bronchopulmonary Aspergillosis (ABPA): This condition occurs in people with asthma or cystic fibrosis and is caused by an allergic reaction to a fungus called Aspergillus. It can lead to coughing, wheezing, and other respiratory symptoms, making it a rare but important consideration in the differential diagnosis.