Wegener's Granulomatosis (Granulomatosis with Polyangiitis, GPA): This condition is a form of vasculitis that affects small- and medium-sized vessels and is known for its involvement of the respiratory tract (causing symptoms like cough and bloody sputum) and kidneys (leading to nephritis and acute kidney injury). The history of epistaxis (nosebleeds) also supports this diagnosis, as it is a common symptom in GPA.

Goodpasture Syndrome: An autoimmune disease characterized by the presence of anti-GBM antibodies, which attack the basement membrane of the lungs and kidneys, leading to hemorrhage in the lungs (causing bloody sputum) and glomerulonephritis (causing acute kidney injury).
Infectious Causes (e.g., Tuberculosis, Pneumonia): Certain infections can cause a productive cough with bloody sputum and, if severe, can lead to acute kidney injury through mechanisms like sepsis or direct infection of the kidneys.
Malignancy (e.g., Lung Cancer): Lung cancer can cause a productive cough with bloody sputum and, if metastasized to the kidneys or causing paraneoplastic syndromes, could lead to acute kidney injury.

Pulmonary Embolism: While less likely to cause a productive cough with bloody sputum for three weeks, pulmonary embolism can lead to acute kidney injury if it causes significant hypoxia or hypotension. Bloody sputum can occur if there is pulmonary infarction.
Endocarditis: Infective endocarditis can cause embolic phenomena leading to renal infarcts (acute kidney injury) and, if the emboli reach the lungs, could potentially cause bloody sputum.
Vasculitis (other forms): Other forms of vasculitis, such as microscopic polyangiitis or eosinophilic granulomatosis with polyangiitis (Churg-Strauss syndrome), could present similarly to GPA and must be considered to avoid missing a potentially treatable condition.

Anti-GBM Disease with Pulmonary-Renal Syndrome: Similar to Goodpasture Syndrome but can occur without the typical anti-GBM antibodies.
Idiopathic Pulmonary Hemosiderosis: A rare condition characterized by recurrent episodes of diffuse alveolar hemorrhage, which could lead to bloody sputum, but it's less commonly associated with acute kidney injury.
Lymphangitic Carcinomatosis: Metastatic cancer to the lungs can cause a productive cough with bloody sputum, but it's less likely to be directly associated with acute kidney injury unless there's significant tumor burden or paraneoplastic syndrome.