  We investigate quantum field theories in two dimensions (2d) with an
underlying Bondi-van der Burgh-Metzner-Sachs (BMS) symmetry augmented by
$\mathfrak{u}(1)$ currents. These field theories are expected to
holographically capture features of charged versions of cosmological solutions
in asymptotically flat 3d spacetimes called Flat Space Cosmologies (FSCs). We
conduct a study of the modular properties of these field theories. The
characters for the highest weight representations of the symmetry algebra are
constructed, and the partition function of the theory is obtained from them. We
derive the density of (primary) states and find the entropy and asymptotic
values of the structure constants exploiting the modular properties of the
partition function and the torus one-point function. The expression for the
asymptotic structure constants shows shifts in the weights and one of the
central terms and an extra phase compared to the earlier results in the
literature for BMS invariant theories without $\mathfrak{u}(1)$ currents
present. We reproduce our field results for the structure constants by a bulk
computation involving a scalar probe in the background of a charged FSC.
