Anglo says it depends on the terms of whatever arrangement governs extradition between Italy and the UK. I don't know whether the European Arrest Warrant ('EAW') would apply, some bilateral treaty or what. In the Shrien Dewani case the English court's hands seem to be pretty much tied by the fact that our treaty with South Africa does not require SA to make out a prima facie case. Dewani's only available ground resistance has been his health. And he is wanted for trial, not to serve a sentence.
Post script (after a little reading):
As was recently
reported here, we have refused to extradite a convicted mafia boss to serve time in Italy because of the state of Italy's prisons. Just reading that link I see it is both the EAW and The Extradition Act 2003 that governed Italy's request, that the requested person would have been extradited but for a decision of the Administrative Court in a case called
Badre -v- Court of Florence. That court in turn considered itself bound to uphold an appeal against an extradition order by a decision of the ECHR (
Torreggiani v Italy 2013) that prison conditions in Italy violated the appellant's Article 3 right not to be subject to inhuman or degrading treatment. The long and the short of it appears to be that, for the time being, until Italy resolves a chronic overcrowding problem, nobody from the UK will be extradited there.
An extract from the Torreggiani case gives the scale of the problem: