According to the book 'New MINI' by Graham Robson the first few pre-production Mini's were built in January 2001 and the first production car on 26th April 2001. The MINI One and Cooper officially went on sale 7th July 2001.
Y96 OBL above was an early press car registered 08 06 2001 and last taxed 30 09 2008. Another one of the early press cars Y286 OBL red Mini was registered 23 05 2001 but was last taxed April 2005 according to DVLA website. (Several photos of it in Graham Robson's book.)
I have seen various spyshots from back in 1999 of some of the disguised new MINI prototypes (registered as Rovers) testing with N & P registrations.
The DVLA website search gives the following details on these first MINI's :-
N735 FYB - Rover MINI - First registered 11 Sep 1998 - 1589cc - White - last taxed until 1 Mar 2000. (see in video below)
N279 NWS - Rover - First registered 11 Sep 1998 - 1589cc - Green - last taxed until 1 Sep 2000. (Export)
N216 FYA - Rover - First registered 21 Jan 1999 - 1588cc - Grey - last taxed until 28 July 2000.
N952 OMR - Rover - First registered 13 Oct 1999 - 1588cc - Black - last taxed until 01 Oct 2000.
P375 YAE - Rover - First registered 13 Sep 1999 - 1588cc - Black - last taxed until 01 Jan 2002.
P489 AHR - Rover - First registered 15 Nov 1999 - 1588cc - Black - last taxed until 01 Nov 2000.
One Rover R50/53 prototype did survive and was sold on ebay a couple of years ago.
^This is said to be the first production MINI No.1 (unregistered)