  The main goal of our paper is to establish a connection between the Weyl
modules of the current Lie superalgebras (twisted and untwisted) attached to
$\mathfrak{osp}(1,2)$ and the nonsymmetric Macdonald polynomials of types
$A_2^{(2)}$ and $A_2^{(2)\dagger}$. We compute the dimensions and construct
bases of the Weyl modules. We also derive explicit formulas for the t=0 and
t=\infty specializations of the nonsymmetric Macdonald polynomials. We show
that the specializations can be described in terms of the Lie superalgebras
action on the Weyl modules.
