how would blackmailers have any better luck than anyone else weaponizing the laws the cops often wont even go after criminals when you hand them video evidence and the criminals location found via tracking device on the stolen property.
Blackmail is about a threat to publish information that could damage reputation — criminal conduct would be such information, especially for an otherwise "upstanding citizen".
If they get as far as actually publishing whatever it was, that means they never got paid.
The UK also allows private prosecution, but that's going to be more of "personal vendettas between non-elites also get prosecuted" rather than "blackmail" given the latter is also a crime.