When a vehicle moves from one base station (BS) to another, a large number of on-board user equipments (UE) may simultaneously and individually perform a handover (HO) procedure, resulting in increased HO overheads. A mobile relay node (MRN), connected via a wireless backhaul to a donor base station (DBS), is deployed on the rooftop of a bus to improve the link quality and reduce the associated HO overhead via group mobility. However, at moderate to high speeds, the on-board UEs can still suffer from frequent HOs due to the MRN failing to HO to a new DBS using the legacy downlink measurement-based HO (DL-HO) method. As a consequence, the connection towards all associated mobile users will be lost which poses tight reliability requirements on the backhaul link to avoid becoming a single point of failure (SPoF). In order to improve the reliability during group handover, in this work, we propose an uplink reference signal (UL RS) based HO procedure (coined as UL-HO) for the MRN which relies on the existing sounding reference signal in long term evolution (LTE) /new radio (NR). In the proposed scheme, and unlike the legacy DL-HO procedure in LTE/NR, the measurement report (MeasReport) transmission is not required between MRN and the DBS, therefore the HO delay can be reduced, decreasing the SPoF chances and thus, uninterrupted services can be provided to on-board UEs. We investigate the gain in terms of HO rate, HO failure rate, ping-pong rate and power consumption (both at the UE and the BS). Performance evaluations demonstrate that the proposed UL-HO scheme outperforms the legacy DL-HO scheme in current cellular networks.