What about in London? If I was going to be visiting London for, say, a week, could I do a ride along with a police officer?
Having worked with London police for some time, I can tell you that, no, you cannot. Until a few years ago this was possible with an advance request, however there was an incident whereby the civilian on the ride-along was injured. To make the long story short, the practice was stopped straight away.
Moreover, even police student officers (i.e. already accepted into the police but still undergoing training) are not normally allowed to go on ride-along, although this rule is a bit easier to bend if you know the right people.
A civilian can take a ride in a police vehicle, as @avid noted, for one of two reasons: if you're arrested (going to the police station) or if you are a victim of crime (drive-around trying to identify the perpetrator). There are also exceptional circumstances where you might end up getting a ride in a police vehicle from point A to point B - but you'd never accompany police officers on their regular duties.
The only exception I know of is for media/journalists working on material about the police - but the formalities to go through to do it are worse than you can imagine.
Only in handcuffs. Or possibly by becoming a victim of crime. Neither are desirable...
However, this might be of interest: http://www.cityoflondon.police.uk/about-us/history/museum/Pages/default.aspx