We study compress-and-forward relaying for delay-universal transmission over a three-terminal network. An important performance measure of delay-universal transmission is the delay-exponent. Two achievable bounds of the delay-exponent are derived. The delay-exponent achieved by a suboptimal message-truncating decoder is inferior to that achieved by an approximate maximum-likelihood (ML) decoder in most cases. However, ML decoding is not analytically tractable when source compression is employed at the relay terminal. Therefore, the system with the message-truncating decoder performs better when signal compression is essential, for e.g. when the source-relay link is non-symmetric and the relay-destination link is noisy.