This paper aims to design and analyze a solution method for a time-dependent, nonlinear and thermally coupled eddy-current problem with a moving conductor on hyper-velocity. We transform the problem into an equivalent coupled system and use the nodal finite element discretization (in space) and the implicit Euler method (in time) for the coupled system. The resulting discrete coupled system is decoupled and implicitly solved by a time step-length iteration method and the Picard iteration. We numerically and theoretically prove that the finite element approximations have the optimal error estimates and both the two iteration methods possess the linear convergence. For the proposed method, numerical stability and accuracy of the approximations can be held even for coarser mesh partitions and larger time steps. We also construct a preconditioner for the discrete operator defined by the linearized bilinear form and show that this preconditioner is uniformly effective. Numerical experiments are done to confirm the theoretical results and illustrate that the proposed method is well behaved in large-scale numerical simulations.