Federated Learning (FL) is a distributed learning scheme that enables deep learning to be applied to sensitive data streams and applications in a privacy-preserving manner. This paper focuses on the use of FL for analyzing smart energy meter data with the aim to achieve comparable accuracy to state-of-the-art methods for load forecasting while ensuring the privacy of individual meter data. We show that with a lightweight fully connected deep neural network, we are able to achieve forecasting accuracy comparable to existing schemes, both at each meter source and at the aggregator, by utilising the FL framework. The use of lightweight models further reduces the energy and resource consumption caused by complex deep-learning models, making this approach ideally suited for deployment across resource-constrained smart meter systems. With our proposed lightweight model, we are able to achieve an overall average load forecasting RMSE of 0.17, with the model having a negligible energy overhead of 50 mWh when performing training and inference on an Arduino Uno platform.