What's the big mystery?
Nissan put together 'CKD' kit ( Completely Knocked Down ) Austins and manufactured Austin-designed cars from late 1952 onwards. The arrangement suited Austin ( they made a decent amount of money out of the deal in a time when it was most welcome ) and it suited Nissan too. The partnership came to an end in 1959.
This is not Clarkson or Top Gear's invention.
He's right you know. American occupation forces were more concentrated on the re-organisation of Japan into a democracy and setting up food distribution. The British Commonwealth Occupational Force was tasked primarily with the decronstrution of the Japanese war industries and their conversion into civil and private companies concentrating on transport infrastructure and vehicles.
America contributed again with the Korean war meaning shipping and the American bases in Japan putting more money into Japan but they still had very little to do with influencing the car manufacturers as much as the British. it was only into the late sixties when the likes of the Bluebird flopped in the states that the Japanese car makers tried more to emulate the larger American vehicles but even then the small coupes and roadsters were scaled on the likes of Triumph etc.