In the near future, the Internet of Things will interconnect billions of devices, forming a vast network where users sporadically transmit short messages through multi-path wireless channels. These channels are characterized by the superposition of a small number of scaled and delayed copies of Dirac spikes. At the receiver, the observed signal is a sum of these convolved signals, and the task is to find the amplitudes, continuous-indexed delays, and transmitted messages from a single signal. This task is inherently ill-posed without additional assumptions on the channel or messages. In this work, we assume the channel exhibits sparsity in the delay domain and that i.i.d. random linear encoding is applied to the messages at the devices. Leveraging these assumptions, we propose a semidefinite programming optimization capable of simultaneously recovering both messages and the delay parameters of the channels from only a single received signal. Our theoretical analysis establishes that the required number of samples at the receiver scales proportionally to the sum-product of sparsity and message length of all users, aligning with the degrees of freedom in the proposed convex optimization framework. Numerical experiments confirm the efficacy of the proposed method in accurately estimating closely-spaced delay parameters and recovering messages.