Clinical scientists are registered with the HPC.
From what I understand the route is via the Grade A training scheme, the entry requirements being a first or upper second class degree.

The traing scheme involves the completion of a masters degree
leads to DipIPEM and is open to graduates with a good honours degree in an appropriate science or engineering discipline, with or without an accredited MSc. Those not already holding an accredited MSc will pursue such a course in Medical Physics or Bioengineering at the University of Surrey, full-time, during the first year of training.

From this I deduce that registered clinical scientists hold a masters degree and have a dedicated training route that can be sponsored by the NHS.