Functions the same as the national network for all intents and purposes, but it is a separate system.
SG would go through to the service controller who would call the signaller if required (thus why some Chiltern drivers will use an SPT instead), as would GSM-R REC. The service controller would then make a separate code red call via Connect if necessary. If a mayday call came via LU Connect first, a code red would follow for Chiltern services via GSM-R shortly afterwards, or the signaller would intervene in other ways.
Similar is true for plenty of other areas where multiple radio networks are used, including the Richmond, Wimbledon and Harrow branches, and on other non-LU networks.