Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally until cooked through but firm to the bite, about 12 minutes. Drain and transfer pasta to a large bowl.
Mix chicken, cream of mushroom soup, and cream of chicken soup into spaghetti. Transfer mixture to a 9x13-inch baking dish and top with Cheddar cheese.
Bake in preheated oven until cheese is melted and sauce is bubbly, about 30 minutes.