Viral Upper Respiratory Infection (URTI): Given the age of the baby, the symptoms of cough and runny nose, and the presence of sick contacts at home, a viral URTI is the most likely diagnosis. These infections are common in infants and are often transmitted through close contact with family members.

Bronchiolitis: This is a common lower respiratory tract infection in infants, often caused by the Respiratory Syncytial Virus (RSV). The symptoms can start with upper respiratory tract symptoms like cough and runny nose before progressing to wheezing and difficulty breathing.
Allergic Rhinitis: Although less common in infants as young as 2 weeks, exposure to allergens could potentially cause symptoms similar to a cold. However, this would be less likely given the short duration of life and the context of sick contacts.

Congenital Heart Disease with Respiratory Symptoms: While not directly suggested by the symptoms of cough and runny nose, infants with undiagnosed congenital heart disease can present with respiratory symptoms due to pulmonary congestion. This is a critical diagnosis not to miss due to its implications for treatment and prognosis.
Bacterial Infections (e.g., Pneumonia, Sepsis): Although the presentation might not strongly suggest a bacterial infection, neonates can rapidly deteriorate from bacterial infections. The absence of fever or more severe symptoms does not rule out these conditions, especially in this age group.
Pertussis (Whooping Cough): Especially if there are known cases in the community or among close contacts, pertussis is a critical diagnosis to consider due to its potential severity in young infants and the need for prompt treatment and isolation.

Cystic Fibrosis: While extremely rare to present with respiratory symptoms this early, cystic fibrosis can cause respiratory issues due to the production of thick, sticky mucus. However, diagnosis typically occurs later in life unless there's a family history prompting early screening.
Primary Ciliary Dyskinesia: A rare genetic disorder affecting the motor function of cilia, leading to chronic respiratory infections. It's unlikely to be diagnosed at 2 weeks of age without other suggestive findings or a family history.