We investigate the coupled propagation of the probe and coherently generated signal fields and the resulting direction-dependent probe-to-signal conversion in a Doppler-broadened four-level closed-loop hot-atom system. By averaging the local linear susceptibilities of different atomic velocity groups in the Maxwell-Boltzmann distribution, we obtain a Doppler-averaged effective-medium response that governs the propagation of the two weak fields. The results show that forward propagation enables a pronounced probe-to-signal conversion, where the signal field gradually builds up along the propagation direction. In contrast, the backward-propagating output is strongly suppressed, resulting in a clear nonreciprocal redistribution of the two-channel outputs. The propagation length, probe detuning, and control-field strengths can all be used to regulate the competition between the probe and signal channels, as well as the probe-to-signal conversion. Mechanistically, the reversal of propagation direction modifies the Doppler-detuning matching for different atomic velocity classes, leading to distinct velocity-averaged responses and thus to conversion behaviors that differ between forward and backward directions. Furthermore, when a vortex phase is imposed on the probe field, the generated signal field still retains the basic radial-mode structure and vortex phase feature of the input field. This work provides a theoretical foundation for implementing nonreciprocal coherent signal conversion, directionally controlled dual-channel outputs, and structured-light-mediated operations in hot-atom systems.