Hello,
I have not started my roundnose Mini for a couple of months. With the key turned off, I connected the battery in the boot. As soon as I connected the battery leads the starter motor engaged, as mentioned the key was in the off location. The vehicle has a push button starter solenoid. Is the solenoid engaged, the button can still be pushed in? How do I turn the car off? The electric fuel pump was not working so the car did not fire?

The starter solenoid may be jammed ON...that's where I would look first...

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"Show me the Mini!"

The ignition switch could also be faulty, but if the electric fuel pump was not operating when the battery was connected, that is about how it should operate as the pump is powered from the ignition switch in the "ON" position. So the ignition switch is probably OK, leaving 9YaTaH's suggestion.
The original Lucas solenoids are getting very old and the reproduction ones seem to be pre-aged.

I had a NEW Solenoid cost $22...DEAD out of the box. We were short of time, so switched to push button start (of the starter) with keyed 12 volts to coil. Bought a spring loaded push button, used a suitable fender washer (bent to provide a mounting lip), turned ignition key to on, pushed button to trigger/hot wire starter. And off we went.

Thanks for your advice. Changed the starter solenoid all working again. Great Help