This should be the top comment. Again and again we see Twitter, Facebook etc. being blamed for this type of stuff. But in truth these companies have well established processes to deal with requests from law enforcement authorities, as do ISPs. This situation is scandalous when you see how much time and effort is devoted to revenue raising through traffic and parking violations, but there is apparently no time to follow up death threats.
"I got on to the authorities again but, polite and sympathetic as they were, there didn't seem much that could be done."
Err... Death threat... Contact Twitter and get the IP address of the user, then contact the ISP to get the home address.
Maybe the police couldn't be "bothered" to do anything, but they were certainly capable of it.