It's a balance of risk, isn't it?
If a setting does a risk assessment, and says 'The risk of covid to these patients, via unvaccinated staff, is unacceptably high', then under H&S legislation, they MUST mitigate that risk to bring it to acceptable levels.
If the staff cannot do their jobs safely (perhaps by using full PPE even once not required for others?) then the other mitigation is a requirement to be vaccinated.