The Department of Health doesn't fund them in England and Wales because it says they are not cost effective, in essence. [Please don't shoot the messenger]. Some considerable while ago there was a post here that referred to a study carried out for the DoH. It was rather heavy going but illuminating, and anyone interested should read it. It is a report by the Medical Care Research Unit of Sheffield University - I have a pdf but cannot remember the web source site.
It is true that the majority of HEMS are provided by one commercial operator. They provide the aircraft, pilot and engineering support. I believe that all paramedics are provided by the local Ambulance Trust/Service, because of a central government edict. The non-paramedic costs of the air ambulance are provided by the local charity - the terms of the contract between the charity and the operator will be the subject of a normal commercial negotiation, but I understand that what jobs they go on are usually determined (or at least requested) by the Ambulance Service Control. The profit motive is there, but control of what jobs are taken on should lie with the charity and Amebulance Service, so I do not see any fundamental problem.
It won't get any votes from the Police, but we tax payers already pay for an emergency service helicopter, the majority of which are under-utilised. Very few police helicopters operate HEMS, which seems to me to be a shame.
Sadly, Cumbria now has neither a Police or HEMS helicopter at present. It seems strange; although Cumbria will have a smallish population, you would have thought that it was ideal country to make good use of a helicpter for the emergency services.
Given some of the crazy things Lottery Fund money gets wasted on, why don't we use some of that for HEMS?