Okay, this isn't a Japanese Wales, just a bit more mixing of Japanese and Welsh cultures.
In the 1980s the Thatcher government uses Wales as a social experiment in untrammeled free markets - businesses there pay no taxes and are very lightly regulated. As a result, every Japanese manufacturing business that wants to sell its products to the European Economic Community sets up in Wales, bringing a lot of wealthy Japanese managers and technicians with them.
Also, in this timeline Ivor the Engine, a 1970s children's cartoon about a train set in Wales, became hugely popular, particularly in Japan. Japanese tourists flock to Wales to travel on steam engines.
Cardiff gets Japanese shops and restaurants.
Japan does not undergo any recession in the 1990s. The bubble of interest in Japanese culture in the west in the late '80's to mid '90s never bursts. Japanese companies in Wales expand as Japan sells more and more stuff to the European Community. Some South Korean companies move to Wales too.
Aardman Animations also moved to Wales in the 1980s. As a result Wallace and Gromit become Welsh. They too become hugely popular in Japan, prompting more tourism. Japanese people start moving to Wales simply because they like it and it's seen as a cool place to live.
Wales is flush with money. The Welsh Independence Party (which has now become more Thatcherite than in OTL) trounces all the other parties in Wales and Wales becomes an independent nation within the European Union. Japanese replaces French as the third language taught in schools (after English and Welsh).