Cancelling before end of the year and no claims bonus

You only get NCD if you hold a policy for a complete year.

One years NCD is worth having ( do dummy quotes online with nil NCD and one years NCD to see if it will be cost effective to keep your policy till the end of the year)

If worth it then consider keeping the policy running

The lack of an MOT won't affect your insurance.

You're correct that technically it's an offence to keep a car with no MOT on the road, but it's a minor offence (small fine and no points) and not one that is routinely enforced. Personally I would probably take the risk of it was only for a few days. If you don't fancy that, is the car driveable and do you have a friend with a driveway where you could keep it for a few days? In fact MOT is one of those requirements which only applies on a road, not a "road or other public place", so it would be legal to keep it in any car park, even a public one.

A third option is simply to scrap it and leave the policy running. There is a risk attached to this that once you've disposed of it, if someone else drives it and doesn't insure it, your own insurer can end up on the hook for any accident they cause, and could then come back to you demanding that you pay the bill because you breached the terms of the policy by not telling them when you sold the car. However if the car is undriveable and obviously fit only for the scrap heap this is probably quite a low risk.