MID is the Motor Insurers Database - set up so that the police can check instantly whether or not a vehicle is insured (rather than wait for a HORT1 to be produced, they can impound the vehicle on the spot)
As the database is populated by insurance companies, they have access to it, and can use it to trace third parties involved in accidents, previous insurers, dual insurers etc.
Get on to your new insurers asap and ask them to do a trace - until they have proof of your no claims bonus, they'll carry on taking the higher amount!