Melt butter in a large pot over medium-high heat. Saute celery and onion in melted butter until tender, about 5 minutes.
Sprinkle flour over the celery mixture; cook and stir until the flour absorbs into the butter, about 2 minutes.
Stream chicken broth, water, and half-and-half into the flour mixture while stirring; add chicken, Cheddar cheese, buffalo wing sauce, and tomato soup. Bring mixture to a simmer and cook, stirring occasionally, until the cheese has melted completely into the liquid, about 10 minutes.
Ladle soup into 8 bowls. Top each serving with blue cheese crumbles to serve.