Rotorbike :
To answer your question, money.
It's been calculated that the WHOLE of the UK could have adequate air ambulance coverage using just 26 strategically placed air ambulances. The money from central or local government just isn't there, because the political will isn't there. The statistics to make a case are perhaps there. For example 46% of air ambulance callouts are to road accidents, a fatal RTA costs the taxpayer in the region of £1million. If the life were saved, would that cost reduce, and if so does it then become economically viable to get into providing air ambulance cover ?
Combining the two services in an area where usage is light allows for increased utilisation of the aircraft, leading over the long term to lower cost to both parties ( police and air ambulance ). Police budgets pay for their share, because charity fund raising isn't seen as an acceptable way to run a police service.
As for new jam sandwiches every time the plates change, the police are different from all the other emergency services since they're primarily used as a proactive, constantly visible service. Fire engines and ambulances don't tour the countryside 24 hours a day touting for business as it were. This means police vehicles tot up the miles quicker than others and therefore need replacement more often. As an example, I was once told by a traffic officer that to do a reasonable job he needed to drive about 120 miles in an 8 hour shift. Each car used like this does 360 miles per day, and if used for say 48 weeks a year clocks up about 121000 miles. Does that help ?