Yep, plugging a device into the OBDII port is a well known cause of a flat battery in the F-Type.
Not all OBDII plug-ins but many, and in the US almost all emissions testing plug-ins cause the problem, it varies from state to state.
What happens is the OBDII plug-in is disconnected / pulled out but it hasn't yet fully shut the session so one or more of the car's electrical modules stay "live" and continue to draw amps, enough so that left overnight like this the battery is flattened.
Reset the Battery Management System (BMS) by disconnecting (unbolting) the earth/ground lead from where it bolts to the floor of the trunk for around 30 seconds then re-connect.
Connect a battery tender/maintainer and leave it going overnight, that should recharge the battery fully. If the battery doesn't recharge fully overnight then it may be beyond salvaging and you would need a new battery.
No point trying to use one of the interior 12v "cigarette lighter" plugs for overnight charging as they all turn off after about 20 minutes.
Heaps of threads, posts and info about this battery problem on the US forum: F-Type ( X152 ) - Jaguar Forums - Jaguar Enthusiasts Forum