Apart from the three languages you named, I know of at least three additional major languages that have used the Chinese script; which are, Thai, Zhuang, and Mongolian. Several minor ones that have also used it include Miao, Yao, Bouyei, Kam, Bai, and Hani.
Thai used to use the Chinese script until the 13th century, when it was abandoned in favor of an Indian Brahmi-family script. Aparently, it used the Chinese characters to mean what they meant in Chinese, but they were pronounced in Thai, in a similar way to Vietnamese.
Zhuang used the Chinese-based characters in the same way, which in Zhuanh is called Sawndip and which means "immature characters". Just like in Vietnamese, Zhuang uses many characters which were created especially for Zhuang, according to the models that the rest of the Chinese characters are built after. The characters which are the same in Chinese and Zhuang mean the same, but in Zhuang they are read using the Zuang words. Sawndip is still used to write Zhuang, along with the official pinyin-style Latin alphabet.
As for Mongolian, the story is markedly different. Mongolian is likely the winner of the title of the language that used to use the greatest number of different writing systems over the course of its history. The most significant book in Mongolian is "The Secret History of the Mongols", written in the 14th century in Chinese characters used solely in their phonetic reading for transcribing the Mongol text, irrespective of their meaning.
Apart from Mongolian, which used the Chinese script to read and write their native languages phonetically, the rest of the languages that used it used to use each character of that script to mean roughly the same it means in Chinese, only they pronounced it the way the concept was named in their language.
There is a rather good Wiki article, "Chinese family of scripts" that can serve as an introduction into this area.