Due to high filler loading, clean, commercial, thermoplastic, flame-retardant materials are mechanically unstable when insulating wires and cables. In this study, composite formulations of linear low-density polyethylene (LLDPE)/ethylene–vinyl acetate (EVA) containing a flame retardant, such as magnesium hydroxide (MH; formula: Mg(OH)2) and huntite hydromagnesite (HH; formula: Mg3Ca(CO3)4, Mg5(CO3)4(OH)2·3H2O), were prepared. The influence of carbon nanotubes (CNTs) and carbon black (CB) on the mechanical properties and flame retardancy of LLDPE/EVA was studied. Three types of CNTs were examined for their compatibility with other materials in clean thermoplastic flame-retardant compositions. The CNTs had the following diameters: 10–15 nm, 40–60 nm, and 60–80 nm. Optimum mechanical flame retardancy and electrical properties were achieved by adding CNTs with an outer diameter of 40–60 nm and a length of fewer than 20 nm. Large-sized CNTs result in poor mechanical characteristics, while smaller-sized CNTs improve the mechanical properties of the composites. CB enhances flame retardancy but deteriorates mechanical properties, particularly elongation at break, in clean, black, thermoplastic, flame-retardant compositions. Obtaining satisfactory compositions that meet both properties, especially formulations passing the V-0 of the UL 94 test with a minimum tensile strength of 9.5 MPa and an elongation at break of 125%, is challenging. When LLDPE was partially substituted with EVA, the limiting oxygen index (LOI) increased. The amount of filler in the formulations determined how it affected flammability. This study also included a reliable method for producing clean, black, thermoplastic, flame-retardant insulating material for wire and cable without sacrificing mechanical properties.