jwf,
When I was studying for my PPLH I had chance to pick the brain of one of the charter pilots who had done Police / HEMS flying.
Positions are sought after due to the nature of the work, especially with HEMS - rather than being a charter pilot flying from a to b, you're part of a team that at the end of the day can save lives. The HEMS flying is varied and involves some interesting landing spots (roads, school fields, etc).
I was told that Police work can also be exciting but can be repetitive at times, often circling a scene, or being stood down at the last minute.
You'll note from the adverts that appear now and then that these positions usually demand pilots with an ATPLH and several hundred or thousand hours, usually on type.
Everyone has to start somewhere and, as with the previous post, getting experience on a turbine machine is vital before the Police / HEMS recruiters will even consider you.
Unless you've got a stack of cash tucked away in the loft or take a military route, then offshore is a good plan - that's the route the pilot I mentioned above took.
Also, as with much in life, it's who you know that helps plus applying when jobs are available.
Hope this helps as it's what I've been advised. If you get an answer from 'the horses mouth' then all the better!
WB