How I'm supposed to diagnose it from a distance christ knows !
Can you plug the ECU into the phone and download it to my workshop ?
Oh it's an old Mini, possibly a condenser (are the points black ?
Could be the coil (I trust you are not running 12 volts into it) as I seem to remember it's got a cold start system on it.
I wouldn't think it's fuel related as you say it starts straight away.
Just bung an old 12v coil on it and juice it up from a 12v feed to see if it runs OK.
Just had a thought, a lot of these replacement coils leak the oil out if they are fitted on their sides or upside down, that will cause this problem.