Although Transformers have gained success in several speech processing tasks like spoken language understanding (SLU) and speech translation (ST), achieving online processing while keeping competitive performance is still essential for real-world interaction. In this paper, we take the first step on streaming SLU and simultaneous ST using a blockwise streaming Transformer, which is based on contextual block processing and blockwise synchronous beam search. Furthermore, we design an automatic speech recognition (ASR)-based intermediate loss regularization for the streaming SLU task to improve the classification performance further. As for the simultaneous ST task, we propose a cross-lingual encoding method, which employs a CTC branch optimized with target language translations. In addition, the CTC translation output is also used to refine the search space with CTC prefix score, achieving joint CTC/attention simultaneous translation for the first time. Experiments for SLU are conducted on FSC and SLURP corpora, while the ST task is evaluated on Fisher-CallHome Spanish and MuST-C En-De corpora. Experimental results show that the blockwise streaming Transformer achieves competitive results compared to offline models, especially with our proposed methods that further yield a 2.4% accuracy gain on the SLU task and a 4.3 BLEU gain on the ST task over streaming baselines.