Pope On A Rope?

The publicity seekers who are trying to get the Pope arrested seem to be making a very strange argument:

Geoffrey Robertson, a UK barrister, and solicitor Mark Stephens believe the Pope would be unable to claim diplomatic immunity from arrest because although his tour is a state visit, he is not the head of a state recognised by the United Nations.

Surely the relevent point is whether the Vatican is recognised as a state by the UK, not the United Nations. Britain has full diplomatic relations with the Vatican, so in order to make the case that the Pope isn't a head of state you also have to argue that the United Kingdom has no say in who it recognises as another sovereign state.

Also the Pope's supposed offences were committed abroad, so whilst simultaneously losing the right to choose who to recognise as a nation we also gain the right to run the legal systems of every other country on the planet, if we follow the internal logic of the likes of Geoffrey Robertson.