This paper investigates energy-efficient transmission protocols in relay-assisted federated learning (FL) setup within industrial subnetworks, considering latency and power constraints. In the subnetworks, devices collaborate to train a global model by transmitting their local models at the edge-enabled primary access (pAP) directly or via secondary access points (sAPs), which act as relays to optimize the training latency. We begin by formulating the energy efficiency problem for our proposed transmission protocol. Given its non-convex nature, we decompose it to minimize computational and transmission energy separately. First, we introduce an algorithm that categorizes devices into single-hop and two-hop groups to decrease transmission energy and then selects associated sAPs. Subsequently, we optimize the transmit power, aiming to maximize energy efficiency. To that end, we propose a Sequential Parametric Convex Approximation (SPCA) method to configure system parameters jointly. Numerical results demonstrate a significant reduction in outage probability and at least a twofold savings in total energy consumption, together with faster convergence, compared with single-hop transmission.