I was offered a temp job doing civilian work at a police station (analysing child abuse reports). The job offer was rescinded when they realised that I had only been in the UK for a few months, and they didn't have time to do a background check.
So this might be a problem if you want to be a police officer.
Also, in New York City, there is a residence requirement for police officers, the idea being that police officers should fit in with and understand the people they are policing. I don't know if parts of the UK have a similar requirement, but its possible they may want you to have lived in the area for a certain amount of time before you can apply to be a police officer.
