The precessing vortex core (PVC) phenomenon in swirling jets is a helical instability in the flow driven by the coherent precession of the vortex breakdown bubble (VBB) around the flow axis, resulting in the helical rollup of the shear layer. This instability is driven by flow processes in the region upstream of the VBB. Centerbodies, commonly employed in combustor nozzles, create a centerbody wake recirculation zone (CWRZ) that can interfere with VBB precession and hence suppress the PVC. We study this phenomenon in a swirl nozzle with a centerbody whose end face is flush with the nozzle exit plane, using large eddy simulations (LES) and linear hydrodynamic stability analysis for flow Reynolds numbers Re = 48,767 and 82,751, based on nozzle exit diameter and bulk flow velocity. For one of the Re = 82,751 cases, the centerbody end face diameter is halved, resulting in the onset of coherent VBB precession. Linear stability analysis reveals a marginally unstable mode in this case. The same mode is found to be stable in the nominal cases. Structural sensitivity analysis shows that the VBB precession eigenmode is sensitive to changes in the time-averaged flow in the VBB-CWRZ merger region. This suggests that the reduction in CWRZ length due to halving the centerbody end face diameter is the reason for the onset of VBB precession. These results suggest that in general, spatial separation between the CWRZ and VBB can result in the onset of VBB precession and the emergence of PVC oscillations in flows with swirl.