Cloud radio access network (C-RAN) is a prominent architecture for fifth generation wireless cellular system that is based on the centralisation of baseband processing for multiple distributed radio units (RUs) at a control unit (CU). In this study, it is proposed to leverage the C-RAN architecture to enable the implementation of direct localisation of the position of mobile devices from the received signals at distributed RUs. With ideal connections between the CU and the RUs, direct localisation is known to outperform traditional indirect localisation, whereby the location of a source is estimated from intermediary parameters estimated at the RUs. However, in a C-RAN system with capacity limited fronthaul links, the advantage of direct localisation may be offset by the distortion caused by the quantisation of the received signal at the RUs. In this study, the performance of direct localisation is studied by accounting for the effect of fronthaul quantisation with or without dithering. An approximate maximum likelihood localisation is developed. Then, the Cramer-Rao bound on the squared position error of direct localisation with quantised observations is derived. Finally, the performance of indirect localisation and direct localisation with or without dithering is compared via numerical results.