To answer your example question directly: English. I don’t think Toyota would ever be written in Japanese on a car, but when talking about the company in writing it would usually be rendered in Japanese.
For model names, it’s more possible that it could be in Japanese, but even then I think it’s rare. Even outside of the realm of global industries like cars you’ll find English words and phrases being used to brand everyday local products all over the place. Japan is far more full of English writing than the Anglosphere is of Japanese…or any other language. When actually talking about these products, though, it’ll usually be written in Japanese (not least because most people in Japan aren’t confident enough in their English ability to use/remember those words and their often unpredictable spellings).
For model names, it’s more possible that it could be in Japanese, but even then I think it’s rare. Even outside of the realm of global industries like cars you’ll find English words and phrases being used to brand everyday local products all over the place. Japan is far more full of English writing than the Anglosphere is of Japanese…or any other language. When actually talking about these products, though, it’ll usually be written in Japanese (not least because most people in Japan aren’t confident enough in their English ability to use/remember those words and their often unpredictable spellings).