Heat oil in a large pot over medium-high heat. Add onion, carrots, celery, mushrooms, salt, and pepper. Cook, stirring occasionally, until the vegetables are softened and the mushrooms have begun to release their liquid, about 6 minutes.
Sprinkle flour over the vegetables; cook, stirring, until the vegetables are coated and beginning to brown, about 1 minute more. Add broth and water; cook, scraping up any browned bits, for 1 minute.
Add wild rice and bring to a boil. Cover, reduce heat to maintain a simmer, and cook, stirring occasionally, for 40 minutes. cook until the rice is tender, about 10 minutes more.
Add chicken, milk, lemon juice, and thyme. Cook, stirring, until warmed through.