Those sections of the Road Traffic Act 1988 (as amended by the RTA 1991) which deal with 'dangerous driving' and 'driving without due care and attention' refer to offences on both 'roads' and 'public places'. Such offences can occur on a supermarket car par (or any access road to it, irrespective of who owns the land).
However, the section of the RTA 1988 which deals with wearing seatbelts was not amended by the RTA 19991; it only refers to 'roads', and not to 'public places'. So you're not required to wear a seatbelt on a supermarket car park. It would seem unlikely that a succesful prosecution could be brought for driving on a private access road without a seatbelt but it would first be necessary to consult the supermarket's title document to find out where the public road ends and the private one begins.
Chris