The increasing demand for efficient resource allocation in mobile networks has catalyzed the exploration of innovative solutions that could enhance the task of real-time cellular traffic prediction. Under these circumstances, federated learning (FL) stands out as a distributed and privacy-preserving solution to foster collaboration among different sites, thus enabling responsive near-the-edge solutions. In this paper, we comprehensively study the potential benefits of FL in telecommunications through a case study on federated traffic forecasting using real-world data from base stations (BSs) in Barcelona (Spain). Our study encompasses relevant aspects within the federated experience, including model aggregation techniques, outlier management, the impact of individual clients, personalized learning, and the integration of exogenous sources of data. The performed evaluation is based on both prediction accuracy and sustainability, thus showcasing the environmental impact of employed FL algorithms in various settings. The findings from our study highlight FL as a promising and robust solution for mobile traffic prediction, emphasizing its twin merits as a privacy-conscious and environmentally sustainable approach, while also demonstrating its capability to overcome data heterogeneity and ensure high-quality predictions, marking a significant stride towards its integration in mobile traffic management systems.