Never.

China uses hànzì. (汉字) Korea uses hangeul. (한글) Thailand uses Thai script. (อักษรไทย;)

In fact, Korea used to use Chinese characters exclusively until hangeul was created in the 15th century. Even then, Chinese characters were still used with great frequency until the 20th century, only fading away within the last 20-30 years. Students used to be required to study ... 1800(?) Chinese characters (don't quote me on that, and if someone wants a source, I'll dig up my book) but these days most Chinese studies are optional and only the older generation has a real grasp of more than the basics of hànzì. They are still used in newspapers or papers when a hangeul word might be unclear (homophones), or for big names (China and Japan are both routinely referred to as 中国 or simply 中, and 日本 or 日)

Now, if you're talking about systems like Hànyǔ Pīnyīn or the McCune-Reischauer/Revised Romanization systems, let's be perfectly clear: they have not (and most likely, never will) replace the writing systems of China or Korea. That said...

At what point were these systems introduced?

China

The first transliteration of Chinese to Latin alphabet was in the 17th century, but this wasn't really a "system" as much as a couple of guys wrote some books using the latin alphabet to approximate Chinese phonetics. In the mid 19th century, a Chinese scholar Song Shu recommended the creation of a syllabary, ala the Japanese kana system.

It was around the same time as Song Shu that the Wade Giles system was created by Thomas Wade in 1858 (and improved by Herbert Giles in 1892). It was in use up until the 1970's in some Chinese-English dictionaries.

There was another system created in the 1930's in Russia (Sin Wenz), but it was never particularly popular and beyond a footnote, there's not much to be said about it.

The most common transliteration today is the Pinyin system. It was developed in the 1950's by the Beijing government and it was built on the existing systems of Gwoyeu Romatzyh (not mentioned), Sin Wenz, and Bopomofo. Bopomofo is a syllabery system very similar to Japanese kana that was discontinued in mainland China, but is still in use in Taiwan. The man who ran the committee to create Pinyin was Zhou Youguang, who is still a live at the young age of 110 years old.

Korea

There's 3 systems for transliterating Korean: McCune-Reischauer, Yale, and Revised.

McCune-Reischauer was created around 1937 and was in use in South Korea until 2000 when it was replaced with the Revised system. M-R uses a system of diacritic marks on the vowels. For example, Sinchon (신촌)and Sincheon (신천), two subway stations in Seoul, would be written as Sinchon and Sinchôn. The MR system was (and, to an extent, still is) used frequently in academics. If you ever read about the Korean War, most books will use the MR system, so Daejeon= Taejeon, Busan = Pusan, Daegu = Taegu, Suwon = Suwôn.

The Yale system is really only used in linguistics, so again, it's barely worth mentioning. One would never encounter it in Korea.

The revised system was introduced in 2000, and it simply removed the diacritic marks and uses a set combination of latin letters for Korean sounds. It's on most street signs, at least in Seoul, but out in the provinces, it's still common to find the MR system in use.

Why are they used?

Tourism. Tokyo, Seoul, and Beijing have all hosted Olympics and that would have been impossible if all of the foreign visitors couldn't read the subway signs, or street signs, or figure out how to get around at all. Hangeul is pretty simple to learn to read, but the logographic systems of Japanese and Chinese would be impossible for a short term visitor to get around with ease.

As I said before, the countries you mentioned all have their own writing systems and they all use them (my knowledge of Thai is limited to the Thai neighborhood near where I live, with everything written in Thai). The only use of the latin alphabet in China, Korea, or Japan is limited to naming street signs or subway stations, or other similar things, and it's only for tourists.

For more information on the romanization of languages, check out any introductory college textbook on said languages. I've got Learning Chinese by Julian K. Wheatley which talks about the Chinese systems a bit. If you're interested in the Korean romanization, see Elementary Korean by Jaehoon Yeon.