A cloud radio access network (C-RAN) is considered where the first hop from the user equipments (UEs) to the basestations (BSs) is modeled by the fading Wyner soft-handoff model. The focus is on mixed-delay constraints where a set of messages (so called "slow" messages) are jointly decoded in the cloud unit (CU), whereas the remaining messages (called "fast" messages) have to be decoded immediately at the BSs. This paper presents inner and outer bounds on the capacity region for such a setup. Moreover, the multiplexing gain region is characterized exactly. The presented results show that for small fronthaul capacity it is beneficial to send both "fast" and "slow" messages. However, when the rate of "fast" messages is already large, then increasing it further, deteriorates the sum-rate of the system. In this regime, the stringent decoding delay on the "fast" messages penalizes the overall performance. Our results indicate that this penalty is larger at moderate SNR than at high SNR and it is also larger for random time-varying fading coefficients than for static ones.