Postnasal Drip (Upper Airway Secretion Syndrome): The patient's symptoms of a chronic cough that started with the onset of cold weather, occasional productivity, and association with allergic triggers (cold) without other respiratory symptoms like wheezing or chest pain, point towards postnasal drip as a likely cause. The absence of fever, smoking history, or symptoms suggestive of GERD further supports this diagnosis.

Allergic Rhinitis: Given the patient's complaint of allergic triggers and the onset of symptoms with cold weather, allergic rhinitis could be contributing to or causing the cough through postnasal drip.
Vasomotor Rhinitis: This condition, characterized by a chronic cough and rhinorrhea without the typical allergic triggers, could be considered, especially if the patient's symptoms are exacerbated by environmental factors.
Sinusitis: Chronic sinusitis could present with a chronic cough, especially if the patient has underlying allergies or anatomical issues that predispose to sinus infections.

Lung Cancer: Although the patient is a non-smoker and relatively young, lung cancer can occur in anyone. A persistent cough, especially if associated with other systemic symptoms (which are not mentioned here), warrants consideration of this diagnosis.
Tuberculosis (TB): TB can present with a chronic cough and is a critical diagnosis not to miss due to its public health implications and the need for early treatment. The absence of fever, night sweats, or weight loss does not rule out TB.
Sarcoidosis: This systemic disease can cause a chronic cough among other symptoms and can be asymptomatic or present with non-specific symptoms, making it a "do not miss" diagnosis due to its potential for serious complications if left untreated.

Bronchiectasis: A condition characterized by permanent dilation of parts of the airways, often presenting with a chronic productive cough. It's less likely given the lack of other symptoms such as recurrent infections or hemoptysis.
Eosinophilic Bronchitis: This condition can cause a chronic cough without wheezing or shortness of breath, similar to asthma but without the variable airflow obstruction. It's a rare cause of chronic cough but should be considered in the differential diagnosis.
Hypersensitivity Pneumonitis: An allergic lung disease caused by a variety of antigens, leading to inflammation of the lung tissue. It could present with a chronic cough among other symptoms, but the patient's lack of exposure history makes this less likely.