If the NHS and police were run properly you could certainly make savings in both by cutting out unnecessary wastage spent on tiers of employees that are not providing the front line services along with cutting back on capital expenditure projects. Plus cutting benefits for those that don't have a disability that would prevent them taking up work. I suspect there's a lot that could be cut at a local level too such as the cost of housing economic migrants. There must be a lot of money wasted by councils for council tax bills to be as high as they are or too many people are getting subsidised council tax bills, leaving an unfair burden upon those that are working in higher paid jobs.
What exactly do you mean by run properly, and domyou have any examples of how they are not and what savings you could make?