My house was broken into and my handbag and contents together with our car keys and car were stolen. Intruder was disturbed therefore took nothing else. According to police front door was key bumped rendering mechanism inoperable. On discovering this and checking we have full comperehensive car insurance but unfortunately due to an oversight we have no house insurance

Car insurance should cover the insured value of the car, as it was stolen... the rest you are SOL on.

Provide a copy of the police report to the claims adjuster.

Your car was stolen. I'm not sure it matters how it was stolen. If your car insurance covers theft, then they should pay.