  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.
