The article concerns the application of the FEM method for the prediction of stress and deformation states in a workpiece during the thread rolling process (TR). The analysis covered a new kinematic variant of the TR process in which the basket of the head rotates and is torque-driven, while the workpiece is stationary and the head with the rollers moves axially relative to the workpiece. The TR process was considered as a geometrical and physical non-linear initial and boundary problem with non-linear, moving, and variable in time and space boundary conditions. The boundary conditions in the contact areas of the tool with the workpiece were unknown. An updated Lagrange (UL) description was used to describe the physical phenomena at a typical incremental step. The states of strain and strain rate were described by non-linear relationships without linearization. New discrete systems of motion and deformation equations of the object in the TR were introduced, which take into account the change in the stiffness of the system during the TR process. This equation was solved by the central differences method (explicit). The material parameters were estimated during tensile tests to determine the characteristics of the C45 steel, and a new semi-empirical method was used to determine the relationship yield stress, effective true strain, and effective true strain rate in the thread rolling process. A modified Cowper–Symonds material model was also used to model the displacement process of the wedge on an elastic/visco-plastic body reflecting the TR process. A non-linear dependency of material hardening module depending on strain and strain rate was introduced. To confirm the plane state of deformation and spatial state of stress, an experimental investigation was carried out. The computer models were validated, and a good convergence of the results was obtained. Applications in the ANSYS/LS-Dyna program were developed to simulate the TR process. The developed applications enable a comprehensive time analysis of the states of displacement, strain, and stress occurring in an object consisting of a workpiece (shaft) and a tool (roller) for the case of a plane strain state and a spatial stress state. Exemplary results of numerical analyzes are presented to explain the influence of the friction coefficient on the condition of the thread quality, and the state of deformations and stresses were shown.