The Hazard Register - what is it really for? My 2c:
Operational risk and HSE risk are usually inseparable. AS/NZS31000 is a good guide/basis (as is 45001). Assessment will always be somewhat subjective (both for likelihood and for consequences). The important bits are multiple people working up and documenting the management/treatment of the risks (who, that, where, when & why) and the assessed residual risk which greatly help informed communication.