I don't know what right this is, but in the UK there's no right to do anything with classified information. Directly the opposite: the official secrets act makes it a crime to share it, distribute it, etc. The only real legal option you have if you find some classified information lying on a train (say) is to hand it to the police.
There's no exemption for journalism and it doesn't matter if you have been 'notified' under the act (that just changes the penalties) or if you are British or not (You're on UK soil, UK law applies). Classified information belonging to foreign allied governments would definitely fall under the OSA.
If that's shocking, ok, but that's the law and has been since the last time the OSA was updated.
I am not a lawyer, but I don't think sharing classified information unredacted would count as a political defence in the UK. I suspect that definition refers to political views and activities that don't break UK laws, e.g. I would expect the US would not be allowed to extradite someone purely for their belief that the US should become a communist state. However if they hacked the computers of th US fed, then they'd be fair game.
The treaty itself is problematic and the US recently refused the Saacoolas extradition. So the next time a British citizen is on the line we should refuse as well. But I also agree: this particular case is about the least troubling extradition ever.