Biscrolling aligned electrospun fiber (AEF) sheets and carbon nanotube (CNT) sheets were fabricated for conductive, functional yarns by a versatile dry composite method. Our biscrolling (twist-based spinning) method is based on spinnable polymer fiber sheets and spinnable CNT sheets unlike the previous biscrolling technique using unspinnable nanopowders and spinnable CNT sheets. The CNT sheet in composite yarns acted as effective electrical wires forming dual Archimedean multilayer rolled-up nanostructures. The weight percent of the electrospun polymer fibers in the composite yarns was over 98%, and the electrical conductivity values of the composite yarns was 3 orders higher than those of other non-conducting polymer/CNT composite fibers which were electrospun from polymer solutions containing similar loading of CNTs. We also demonstrate that biscrolled yarns having various structures can be fabricated from spinnable AEF sheets and spinnable CNT sheets.