In Australia (specifically Victoria) this type of work is not the exclusive remit of EMS or public service helicopters.
I had a long standing arrangement with our trauma management to transport organs and even organ teams as required, day or night, and was paid for the service by the hospital. Obviously slightly different to the UK since distances are significant issues, generally harvested organs would be brought in an esky by a scheduled domestic airline and we would meet them on the ramp and then fly to the designated hospital. Organ teams would be taken to a regional hospital to gather the donated organs (a long job, obviously) and we'd then return the team plus harvested organs.
I got involved because of the very reason discussed here, the suitable use of our local VicPol helicopters and EMS machines. Not only did such work take them away from their core tasking but the inter-agency charges were actually quite a lot more than my commercial rates.