The Solicitor General reports that there is no power to appeal against the bizarre result of the High Court's discharge of the Dudley killer without conviction.
 

When name suppression and discharge without conviction combine, the community is doubly insulted by the lawyers' sytem. The discharge without conviction officially says ‘nothing to see here, we'll keep no record that anything legally wrongful happened’. On its own, that is often a consequence of proper legal process – decisions not to charge,acquittals and innocence until conviction combine. .

But when name suppression is added we lose our normal freedom to form and act on our own views on the behaviour, and to remember it for future dealings with the offenders,  whether or not the high criminal standard of proof for exercise of the state's coercive power was reached..

The combination of discharge with name suppression, however, means that it could have been better for the Dudley family if the Police had simply declined to take any action. In that situation their freedom of expression would not have been trampled.
Because the name suppression survives the community can't now officially know who the culprits are, and take any apparent lack of remorse into account in their futures. Of course in their community, where it really matters at present, everyone already knows. But our judges have resolutely declined to consider such realities.

I have strong views about judges discharging without conviction – to me it is airbrushing history, deliberately depriving the community of information vital for healthy functioning of the social, employment and other deterrents that uphold good behaviour and deter bad ..
Our judges collectively ignore the fact that there will never be enough police, judges, prisons and social workers for officialdom to create a society in which people routinely prefer not to offend and the strong are not contemptuous of the weak. Shame and the other social results of offending are vital in all healthy cultures. Behaviour that is not unlawful but is nevertheless reprehensible, must attract its natural costs.
The bleeding hearts should be on our side in this. When the official justice system interferes with shame there is inevitable pressure to crank up ever more severe formal sanctions.

Name suppression after a discharge without conviction is judicial arrogance. It insults the victim’s family. It steals our rights to know, the purpose of freedom of expression. .It undermines the primary social mechanisms, which should be the first and inevitable outcome of wrong-doing. The criminal justice system should be the back-stop, not the first and only social response.

This is not to criticise the Solicitor General. It appears he was left with no power to appeal.