Electrospinning is a well-recognized technique for producing nanostructured fibers capable of supporting cell adhesion and further proliferation. Here, we prepared a novel electrospun blend from poly(butylene adipate-co-terephthalate) (PBAT), a non-conductive and biodegradable polymer, and a conductive polymer, namely polypyrrole (PPy). Therefore, the goal was to create electrically conductive nanoscaffolds for tissue engineering applications. Furthermore, to improve the scaffold biomimetic features for bone regeneration purposes, we demonstrated the feasibility of electrodepositing nanohydroxyapatite (nHAp) onto the new hybrid scaffold. Electrochemical measurements confirmed the electrical conductivity of the novel PBAT/PPy scaffold, which allowed nHAp electrodeposition, further confirmed via ATR-FTIR analysis and FE-SEM micrographs. The PPy loading did not change the fibers' average diameter, although the increase in the solution conductivity was probably responsible for leading to electrospun mats with smaller beads and a lower presence of flattened regions compared to PBAT neat. The hybrid scaffold was more hydrophilic than PBAT neat. The first presented an advanced contact angle (ACA) of 84°, whilst the latter presented an ACA of 115°. The incorporation of PPy to PBAT maintained the ability of the generated scaffold to support cell adhesion with no changes in MG-63 cell viability. However, the PBAT/PPy scaffold presented higher values of alkaline phosphatase, an important indicator of osteoblasts differentiation. In conclusion, we demonstrated a feasible approach to create electrically conductive nanoscaffolds, which are capable of undergoing nHAp electrodeposition in order to generate materials that are more hydrophilic with improved cell differentiation. These results show the potential of application of this novel scaffold towards bone regenerative medicine.