Simultaneous speech translation (SimulST) produces translations incrementally while processing partial speech input. Although large language models (LLMs) have showcased strong capabilities in offline translation tasks, applying them to SimulST poses notable challenges. Existing LLM-based SimulST approaches either incur significant computational overhead due to repeated encoding of bidirectional speech encoder, or they depend on a fixed read/write policy, limiting the efficiency and performance. In this work, we introduce Efficient and Adaptive Simultaneous Speech Translation (EASiST) with fully unidirectional architecture, including both speech encoder and LLM. EASiST includes a multi-latency data curation strategy to generate semantically aligned SimulST training samples and redefines SimulST as an interleaved generation task with explicit read/write tokens. To facilitate adaptive inference, we incorporate a lightweight policy head that dynamically predicts read/write actions. Additionally, we employ a multi-stage training strategy to align speech-text modalities and optimize both translation and policy behavior. Experiments on the MuST-C En$\rightarrow$De and En$\rightarrow$Es datasets demonstrate that EASiST offers superior latency-quality trade-offs compared to several strong baselines.