Would a private police force have bothered to arrest him? Oh - hang on a second .... this was the British Transport Police, funded almost exclusively by the train operating companies , Network Rail and London Underground and with a remit limited to railway land (which suggests either that the hotel was under a railway freehold, or the officers arrested Care as private citizens). So it was indeed a private police force that arrested him ...
British Transport Police (and Heddlu Trafnidiaeth Prydeinig) are set up and empowered by the Home Office in the same way as the rest of the British Police Forces at County, Metropolitan or City levels. As you say that is restricted to relevant land, but so are the other forces constrained by geographical areas. They are paid for by the contracting transport companies, as are county forces by their Police Authority.
Ulster is different as there is just one Police Force, who cover transport as well.
It is not a private police force legally - as you well know you "stirrer" you :biggrin:
In certain circumstances a BTP constable can act as a police constable outside of their normal railway jurisdiction as described in the "Powers and status of officers" section of their charter. Under "spontaneous requirement outside natural jurisdiction" a BTP constable has the same powers and privileges of a constable of a territorial police force !