This happened to a friend of mine He reported it but still had police at his door at 11 one evening he answered door 1/2 cut police asked if he was keeper of XXXXXXX and could they see it His car had a distinctive grill not the same car as used but police still had to check as Some unscrupulouse peeps reported theft of plates then did their own petrol fill n run Saying "it must have been the stolen plates" when police asked only to be caught out due to their car having non standard alloys
tip from the police was for him to carry docs with him when in the car as they WOULD stop it to confirm
i had the same thing happen to me many years ago, i had an xr3i back in the day, i used to live in mid wales, after finishing work in the early hours, i drove home, nose of the car was under a shelter ( based in an old railway station

), got up the next morning, bumper inc no plate, bodykit & spot lamps gone

reported to the police, in surrey 2 wekks later visiting friends, police car behind me, thought nothing of it, got to a junction just uotside purley, police car's blocked the road, one behind boxed me in, as no plate reported stolen in wales, had all doc's with me, still had untold grief trying to convince them it was mt car, & i had repored them stolen!!!!