Apart from the cars he modified for his own racing team in the late 60s/early 70s John Cooper did not make any of the Minis that bear his name until 1995 when he modified standard factory Mini Coopers as special editions with high performance conversions. There were no British marketed Coopers between 1972 and 1989.
In the late 80s he sold engine conversion kits for 998 Minis in Japan and by 1989 he had agreement to sell them in the UK with Rover warranty. The most obvious visible feature is that it has twin carbs. I can see no reason why an automatic Mini might not have had the conversion, though not perhaps the best application.
The Mini Cooper was officially reintroduced by Rover in 1990 with a 1275 engine in a limited edition (now called RSP) for the UK and Japan markets. John Cooper introduced a new 'S' conversion kit for the 1275 engine in 1991.
The Mini became very popular in Japan long after the original Coopers were discontinued. This was the source of much disappointment and one of the reasons why the Cooper was revived in 1990. Apparently it was very common for ordinary 80s Minis to be turned into Cooper-lookalikes in Japan.