I was involved over 20 years ago, so my experiences are now pretty dated. Back then, different police forces had different approaches, so many had established breeding programmes with known lines bred for temperament, trainability etc. But some (like Devon & Cornwall back then) used a combination of being offered pups, identifying candidates from rescue centres and increasingly using breeding programmes. They all seem to use in-house breeding programmes now, it seems unusual (but not unheard of) for grown dogs to be taken on. 
As I remember, the pups that we looked after were brought to their families at around 12 weeks old, and we were given very clear instructions and advice on socialising; food, toys etc. was all supplied. The aim was to give them 'normal' upbringings so that their little characters could emerge at their own pace - it was all very light touch, I'm not sure if that's still the case. We had regular visits from dog handlers to check on progress, and the pups would have regular assessment days at HQ which would decide whether they continued on the course or were released to families (there was always a long waiting list of ex-handlers, people who'd puppy walked previously and others keen to take them). At 10 months old there was a full assessment, and another at 12 months when I think they were assigned to handlers who then took them on to live with them. we were very fortunate that the handlers of pups we looked after kept in touch. I think they generally retired at 7 unless injured, and then were either rehomed or more often than not stayed with their handler as a retired police dog.
The main thing that struck me was how much the handlers adored their dogs, and welfare was absolutely top priority. The bonds they developed were phenomenal, we heard many stories of 'our' dogs (and others) doing incredible things for their handlers.