the UK would never switch to a government led AA model
Pretty sure Scotland is still in the UK, and the HEMS work up there is shared by the government-funded Air Ambulance and the Scottish Charity Air Ambulance. I haven't operated up there since 2015 so I don't know if both are also involved in routine/emergency hospital transfers too.
Some charities work exceptionally well - the RNLI (in my opinion) is the gold standard that we in this country should aspire to for all emergency-related services. Don't know enough about the Oz RFDS to know if it is a good exemplar too.