This study addresses the achievable sum-rate for the uplink of a cloud radio access network (C-RAN) operating in a linear Wyner-type topology, i.e., with partial channel connectivity. In the system, the radio units (RUs) communicate with a central, or cloud, unit (CU) by means of digital finite-capacity fronthaul links. The messages sent by the user equipments (UEs) are jointly decoded by the CU based on the compressed baseband signals received on the fronthaul links. Unlike prior works, each RU is assumed to be also connected to its neighboring RUs via finite-capacity fronthaul links. Under the standard assumption that the RUs do not perform channel decoding (i.e., oblivious RUs), each RU performs in-network processing of the uplink received signal and of the compressed baseband signal received from the adjacent RU, with the CU carrying out channel decoding. A closed-form expression of the achievable sum-rate is derived assuming point-to-point compression, and then analytical expressions are provided for more advanced fronthaul compression schemes that leverage side information. Numerical examples provide insights into the advantages of inter-RU communications and into the performance gap to existing sum-rate upper bounds.