The wording in bold below is from the Churchill policy general conditions page 15.
1 Your duty
We will only provide the cover set out in this policy if you keep to all the terms and conditions of the policy.
6 Taking care of your vehicle
make sure your vehicle is roadworthy;
take all reasonable steps to protect your vehicle and its contents from loss or damage;
make sure you keep property left in an open or convertible car in a locked boot or locked glove compartment; and allow us to examine your vehicle at any reasonable time if we ask you.
Churchill should accept your explanation as to why the vehicle had no MOT at the time of the theft. Usually Insurers will only decline a claim if the breach of policy condition was relevant to the loss event. If the car was securely locked and keys removed at the time it was stolen from your driveway then I could see no reason for claim being declined.
As Dacouch has commented, not having an MOT may lead Churchill to only paying the trade value. Insurers tend to uses glasses guide to assess market value but will also take into account any information you can supply them e.g. service record. See glasses guide link
Dacouch is also right that if the car has a relatively low value, when taking into account the excess and increased premium, particularly if no claims discount is reduced, this can make it not worth claiming. It is worth looking into this, if this is relevant.
I am sure Churchill may ask more questions due to lack of MOT. Unfortunately, this is a sign of the times, with Insurers experiencing a growing number of fraudulent claims. Don't be put off by this, just provide any information they are seeking when they request it.