  We describe the general strategy for lifting the Wess-Zumino-Witten model
from the level of one-loop Kac-Moody $U_q(\widehat{\mathfrak{g}})_k$ to generic
quantum toroidal algebras. A nearly exhaustive presentation is given for the
two series $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$ and
$U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_n)$, when screenings do not exist
and thus all the correlators are purely algebraic, i.e. do not include
additional hypergeometric type integrations/summations. Generalizing the
construction of the intertwiner (refined topological vertex) of the
Ding-Iohara-Miki (DIM) algebra, we obtain the intertwining operators of the
Fock representations of the quantum toroidal algebra of type $A_n$. The
correlation functions of these operators satisfy the
$(q,t)$-Knizhnik-Zamolodchikov (KZ) equation, which features the ${\cal
R}$-matrix. Matching with the Nekrasov function for the instanton counting on
the ALE space is worked out explicitly. We also present an important
application of the DIM formalism to the study of $6d$ gauge theories described
by the double elliptic integrable systems. We show that the modular and
periodicity properties of the gauge theories are neatly explained by the
network matrix models providing solutions to the elliptic $(q,t)$-KZ equations.
