Hi Jon what's probably happened is your aircon compressor has seized (common fault ) for some reason the aircon often switches itself on by itself in cold weather refrigerant can remain liquid and enter the compressor as you can't compress a liquid the compressor "hydraulics " and destroys itself inside new one is damm expensive and system needs cleaning out well You can do without the aircon till April so just get the car running . First thing to try is disconnect the plug close to the compressor it's a bit fiddly to get at so another way is takeout the mini relay in the power distribution box next the battery Now start the engine look down at the compressor pulley you should see the pulley turning and the centre hub should sit still If it shows any sign of trying to move or starts to get hot the clutch and pulley are seizing In this case you could try this loosen the bolt on the centre of the compressor hub prise off the end cap now you can see if the clutch is seized Also check this , release the drive belt ( since you saw it smoking inspect it is not damaged ) check the compressor hub is not wobbly you should be able to spin the pulley part nice and free Good luck
