It may be nothing but I would get under the bonnet and make sure all your battery connections are clean and tight along with connections on starter. A dirty or loose connection will cause resistance and in turn heat which can start to melt the cable insulation causing the smell you describe.
Check your drive belt for the supercharger/alternator that could be slipping or getting hot with the extra alternator load after starting.
Worth checking the steering pump isn't on its way they can cause electrical issues
One thing I would say is are you sure it's an electrical plastic smell. Cooper s and come to think of it cooper cars are prone to leaking coolant here and there. This can be a bit smelly on start up. Check your expansion tank they leak around the seam half way up the body of the tank sometimes. You will see a white or blue residue where it has run down the plastic tank from the seem. Also I had a split heater matrix that stunk the car out and steamed the window up!
Other than that check all fuses are in place correctly and the right rating, make sure they have a good contact.
Hope that helps I'm sure others will think of some things I have missed. Good luck and let us know how you get on.