The noise that i think you are on about is the hydraulic tappets. It must be.
One the Mini's the tappets which controll the valves are hydraulic if you did not know. When the car has been stood for a while, the oil in the engine due to gravity slowly makes its way back to the sump.
So when you start up the engine, there is not allot of oil surrounding the tappets, when the oil pump pumps the oil to the designated area, the noise will go.
I hope this helps you.
Mine does exaclty the same thing, if it is the same noise that is.
Regards
Magik ( Ben )
