Depending on how they designed the network setup, it could be either way.
If they setup so that only a handshake is required and then you transfer to a "host" session, then the host and anyone joining will have a period of ISP traffic. If the host and client are local, then it could send the traffic directly over the local network - after all, the host is the server and not on the internet.
Any session change would require a server update. So if you stay in the same region of space then it's all handled locally. If anyone wants to join your session, or one of you goes outwith your "zone", or some random event kicks off then there should be some form of ISP traffic between your host server and the one in the UK and then the clients.
Having everyone on the same LAN would give you an advantage, your client responses would be faster as latency really wouldn't play a part.
Of course everything could all be driven from one central server, in which case the more folk you have on the same LAN = the slower the ping!