The deformed $mathcal W$ algebras of type $textsf{A}$ have a uniform description in terms of the quantum toroidal $mathfrak{gl}_1$ algebra $mathcal E$. We introduce a comodule algebra $mathcal K$ over $mathcal E$ which gives a uniform construction of basic deformed $mathcal W$ currents and screening operators in types $textsf{B},textsf{C},textsf{D}$ including twisted and supersymmetric cases. We show that a completion of algebra $mathcal K$ contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except $textsf{D}^{(2)}_{ell+1}$. We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.
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