Buy Now Pay Later (BNPL) is a rapidly proliferating e-commerce model, offering consumers to get the product immediately and defer payments. Meanwhile, emerging blockchain technologies endow BNPL platforms with digital currency transactions, allowing BNPL platforms to integrate with digital wallets. However, the transparency of transactions causes critical privacy concerns because malicious participants may derive consumers' financial statuses from on-chain asynchronous payments. Furthermore, the newly created transactions for deferred payments introduce additional time overhead, which weakens the scalability of BNPL services. To address these issues, we propose an efficient and privacy-preserving blockchain-based asynchronous payment scheme (Epass), which has promising scalability while protecting the privacy of on-chain consumer transactions. Specifically, Epass leverages locally verifiable signatures to guarantee the privacy of consumer transactions against malicious acts. Then, a privacy-preserving asynchronous payment protocol is further constructed by leveraging time-release encryption to control trapdoors of the redactable blockchain, reducing time overhead by modifying transactions for deferred payment. We give formal definitions and security models, generic structures, and formal proofs for Epass. Extensive comparisons and experimental analysis show that Epass achieves KB-level communication costs, and reduces time overhead by more than four times in comparisons with locally verifiable signatures and Go-Ethereum private test networks.