While FL is a widely popular distributed ML strategy that protects data privacy, time-varying wireless network parameters and heterogeneous system configurations of the wireless device pose significant challenges. Although the limited radio and computational resources of the network and the clients, respectively, are widely acknowledged, two critical yet often ignored aspects are (a) wireless devices can only dedicate a small chunk of their limited storage for the FL task and (b) new training samples may arrive in an online manner in many practical wireless applications. Therefore, we propose a new FL algorithm called OSAFL, specifically designed to learn tasks relevant to wireless applications under these practical considerations. Since it has long been proven that under extreme resource constraints, clients may perform an arbitrary number of local training steps, which may lead to client drift under statistically heterogeneous data distributions, we leverage normalized gradient similarities and exploit weighting clients' updates based on optimized scores that facilitate the convergence rate of the proposed OSAFL algorithm. Our extensive simulation results on two different tasks -- each with three different datasets -- with four popular ML models validate the effectiveness of OSAFL compared to six existing state-of-the-art FL baselines.