Derfred,
From the AIP
Minimum Fuel: The term used to describe a situation when an aircraft's fuel supply has reached a state where having committed to land at a specific aerodrome, the pilot calculates that any change to the existing clearance to that aerodrome may result in landing with less than fixed fuel reserve for the flight.
Note: The minimum fuel state is not an emergency situation but an indication that an emergency situation is possible should any additional delay occur.
After declaring "Minimum Fuel" I doubt that the response will be "state your diversion airport" as by definition you have supposedly committed yourself to land at the destination.