With the wide deployment of electronic sensor networks, the capability of extracting meaningful knowledge from nonstationary data streams has become increasingly important in real-world applications. Boosting has been proven effective in constructing highly accurate predictive ensemble models in offline settings but remains underexplored in online application scenarios. Meanwhile, evolving fuzzy systems (EFSs) have been widely recognised as powerful tools for learning from nonstationary data streams thanks to their real-time adaptability, high transparency, and model interpretability, which are particularly valuable in high-stakes applications. Despite these advantages, the use of EFSs as base classifiers within online boosting frameworks has not yet been investigated. In this paper, a novel online fuzzily weighted adaptive boosting algorithm (OFWAB) is proposed to enhance the performance of first-order EFSs in data stream environments. OFWAB sequentially trains a series of EFSs on a sample-by-sample basis and dynamically adjusts the sample weights and classifier weights according to the confidence in their respective predictions, effectively adapting to the evolving characteristics of streaming data. To further improve learning under limited supervision, a semi-supervised extension, S²OFWAB, is introduced. S²OFWAB enables the ensemble classifier to continuously self-improve from unlabelled streaming data via pseudo-labelling. Extensive numerical experiments on a variety of benchmark datasets under both online supervised and semi-supervised settings demonstrate that OFWAB and S²OFWAB effectively boost the performance of the underlying EFSs, outperforming state-of-the-art methods in terms of accuracy and adaptability to nonstationary data. This work presents a promising direction for building self-evolving online ensemble classifiers for real-time data stream applications.