I have had a number of experieces of a different kind with the police.
Firstly, somewhat amusingly, a policeman reversed into me. He was queueing up at a roundabout near the police station, and the car in front was stuck, and he reversed without looking! Fortunately it was at low speed, so no damage caused. I did ask to see his warrant card though to make him feel even more uncomfortable!
Now two experiences associated with my former work with the NHS. I worked with them to establish three rape havens in London. What struck me was, that despite the service being 50:50 funded by the police and NHS, the NHS had 5 attendees representing the three havens and central NHS, whereas the police had probably around 12 - 15 attendees, mostly uniformed guys and lasses. It all seemed very bureaucratic (including decisions having to be approved by 'upstairs', and I have to say that that was not the case with the NHS side - we were able to make all the necessary decisions), but having said that, they were all very helpful and good people to work with.
Secondly, the NHS and other public bodies (fire, armed services, councils, coastguard etc) all work together for any major incidents, such as flooding, major accidents etc. The police usually take the lead on this, and are very good at managing the incidents. There are regular meetings (often hourly) throughout the day and night, chaired by a senior police officer, and it is all very closely controlled and managed. A clear case of command and control in action. It does seem to be bureaucratic, but, given that the actions could be reviewed at a Public Inquiry, all actions, and reasons for them, need to be documented. From my experience, the police do excel at this, and do manage what, for the staff in the other agencies involved, can be a very stressful experience. I am an accountant by training, but still was on-call for such incidents, even though it was outside my usual line of business, so was grateful for the police being so well organised to run such incidents.