That's what I thought about heritage railways too. It's private land. Surely a ticketing issue or whatever (within reason) would be a civil matter.
Not necessarily. The KWVR, for example, have Byelaws which are almost word for word the same as the national network, thus making such matters a criminal matter rather than civil.
In respect of the NYMR, I believe they are policed by BTP, and have been for a few years, although not since they started operating on the national network, I think it was a more recent change.
I know through my work on the national network that when there has been a matter requiring urgent police response that if the BTP cannot immediately attend, they will contact the local constabulary and request assistance from them. Having been requested by another force to assist, they then do so, normally fairly promptly. My gut feeling is that those local forces would give higher priority to requests from another force than they might if it were a member of public / staff ringing them directly and asking for assistance. The BTP would have a better overally understanding and picture of railway operations and matters than a local force and devise an appropriate response.
That said, the KWVR has in recent times always got a fairly prompt response from the West Yorkshire Constabulary in respect of matters of trespass, etc. My only concern is that because they lack the requisite training or specific railway knowledge of BTP, some of their decision making in how to deal with a matter is not as helpful as BTP - for example wanting a complete line closure or blockage before they will do something, rather than a more measured proceed at caution type of response.