Allergic Rhinitis: This is the most likely diagnosis given the chronic nature of the symptoms (12 days without improvement) and the absence of fever. Allergic rhinitis often presents with nasal discharge and cough due to postnasal drip, and it is common in school-age children.

Viral Upper Respiratory Infection (URI): Although the duration is a bit long for a typical URI, some viruses can cause prolonged symptoms. The lack of fever and the specific symptoms of nasal discharge and daytime cough could still fit within the spectrum of a viral URI.
Sinusitis: This could be a consideration, especially if the nasal discharge is purulent or if there are other symptoms such as facial pain or headache. However, the absence of fever makes acute bacterial sinusitis less likely, though chronic sinusitis could be a possibility.
Asthma: Cough can be a predominant symptom of asthma, especially if it is exercise-induced or triggered by allergens. The chronic nature of the cough and the absence of other symptoms like wheezing or shortness of breath do not rule out asthma, especially in a child who may have allergic triggers.

Pertussis (Whooping Cough): Although less common due to vaccination, pertussis can present with a chronic cough and is a critical diagnosis not to miss due to its contagiousness and potential severity, especially in infants who may be exposed to the child.
Tuberculosis (TB): A chronic cough in a child could be a sign of TB, especially if there has been exposure to someone with TB. This diagnosis is crucial not to miss due to its public health implications and the need for early treatment.
Foreign Body Aspiration: If the onset of symptoms was sudden, a foreign body in the airway could be a possibility, although this is less likely given the duration of symptoms and the absence of other signs like choking or acute respiratory distress at onset.

Cystic Fibrosis: This genetic disorder can cause chronic respiratory symptoms, including cough and nasal discharge, due to the production of thick, sticky mucus. It's a rare condition but important to consider in a child with persistent respiratory symptoms.
Primary Ciliary Dyskinesia (Kartagener Syndrome): A rare genetic disorder that affects the cilia in the respiratory tract, leading to chronic respiratory infections and symptoms like cough and nasal discharge.
Gastroesophageal Reflux Disease (GERD): Can cause a chronic cough due to acid reflux irritating the airways. Though more common in infants, it can occur in school-age children, especially if they have other symptoms like heartburn or abdominal pain.