My understanding is that 'British Rescue' callsign only applies to SAR flights when tasked by JRCC. Not HEMS.
From
Policy Statement (caa.co.uk)
SAR and Coastguard flights.
Bristow Group operate Coastguard and SAR operations in the UK under contract to the British Government. It should be noted that “COASTGUARD” callsigns can be re-tasked as SAR flights by the Joint Rescue Coordination Centre (JRCC); these flights will then adopt the call sign ‘BRITISH RESCUE’.
Air traffic services (ATS) units in communication with the aircraft at the time of the re-tasking should inform the ATS unit, where one exists, at the departure aerodrome, of the change in callsign
HEMS and Police callsigns remain as they are today.