An asynchronous $\ka$-active-user unsourced multiple access channel (AUMAC) is a key model for uncoordinated massive access in future networks. We focus on a scenario where each transmission is subject to the maximal delay constraint ($\dm$), and the precise delay of each user is unknown at the receiver. The combined effects of asynchronicity and uncertain delays require analysis over all possible delay-codeword combinations, making the complexity of the analysis grow with $\dm$ and $\ka$ exponentially. To overcome the complexity, we employ a wrap-decoder for the AUMAC and derive a uniform upper bound on the per-user probability of error (PUPE). The numerical result shows the trade-off between energy per bit and the number of active users under various delay constraints. Furthermore, in our considered AUMAC, decoding without explicit delay information is shown to achieve nearly the same energy efficiency as decoding with perfect delay knowledge.