We investigate the physics of Ionic Liquids (ILs) doped with lithium salts in both bulk and when confined within vertically aligned Carbon NanoTubes (CNT) composite membranes. Recognized for their exceptional chemical and electrochemical stability, ILs have emerged as promising electrolytes for the development of secure and sustainable energy storage systems. When confined in a macroscopic 1D CNT scenario, we demonstrate a significant one-order-of-magnitude enhancement in the ionic conductivity of IL-based electrolytes. This suggests that CNT membranes offer a potential avenue to amplify Li+ transport properties and consequently increase the specific power of solid-state batteries. In this context, we scrutinize the multiscale dynamics of IL-based electrolytes in both bulk and 1D CNT confinement, employing neutron scattering techniques (QENS, NSE, and backscattering at the molecular scale), PFG-NMR (at the micrometer scale), and electrochemical measurements (at the macroscopic scale). At the molecular scale, under confinement, the dynamics are activated at lower temperatures (between 10 and 20°C) compared to the bulk, accompanied by an increase in the long-range translational diffusion coefficient. Molecular dynamics simulations results allow us to attribute the conductivity improvement to a reorganization of the electrolyte's nanostructure under confinement.