Mobile service robots are proving to be increasingly effective in a range of applications, such as healthcare, monitoring Activities of Daily Living (ADL), and facilitating Ambient Assisted Living (AAL). These robots heavily rely on Human Action Recognition (HAR) to interpret human actions and intentions. However, for HAR to function effectively on service robots, it requires prior knowledge of human presence (human detection) and identification of individuals to monitor (human tracking). In this work, we propose an end-to-end pipeline that encompasses the entire process, starting from human detection and tracking, leading to action recognition. The pipeline is designed to operate in near real-time while ensuring all stages of processing are performed on the edge, reducing the need for centralised computation. To identify the most suitable models for our mobile robot, we conducted a series of experiments comparing state-of-the-art solutions based on both their detection performance and efficiency. To evaluate the effectiveness of our proposed pipeline, we proposed a dataset comprising daily household activities. By presenting our findings and analysing the results, we demonstrate the efficacy of our approach in enabling mobile robots to understand and respond to human behaviour in real-world scenarios relying mainly on the data from their RGB cameras.