Good question - it is perhaps worth noticing that private prosecutions are rather rare and typically only happen when the "official" prosecutor appears to have made an error of judgement and appear to be taken by victims (or their families) as an act of desperation.
What happens in other countries if the "official" prosecutor chooses not to prosecute a crime?
NB I'm not intending to imply that I think the various approaches to private prosecutions that we have in the UK are a good thing - this case brought by FACT does look a bit worrying.
In the US, if the prosecutor with authority over the offense chooses not to prosecute, and no other venue exists to prosecute the offense (for instance, there's no federal angle to the offense that would get the FBI involved), that's where the matter ends. And that's exactly how things do end, all the time.
This is mostly a good thing. If anything, we need more discretion over charging cases.
For example:
http://www.firmmagazine.com/features/155/When_worlds_collide...
http://en.wikipedia.org/wiki/Murder_of_Stephen_Lawrence
What happens in other countries if the "official" prosecutor chooses not to prosecute a crime?
NB I'm not intending to imply that I think the various approaches to private prosecutions that we have in the UK are a good thing - this case brought by FACT does look a bit worrying.