India and Korea did not adopt Western Abjads and Alphabets into their own writing systems. Let's uh, put the kaibash on that right away. Indian writing systems, including Devanagari which a modernized version is currently used to write Hindi and Nepali, and has been one of the primary scripts for Sanskrit, are Brahmic, mostly descending from the Brahmi script that were used to write Ashoka's pillar edicts. Korea today uses Hangul, a syllabic featural alphabet that was intentionally designed for the Korean tongue.

The Korean story is not my forte, but I can summarize it's essential story: Sejong Daewang, or "Great King Sejong" was the fourth king of the Joseon Yi Dynasty and developed a "Hall of Worthies," giving these scholars a series of tasks, including developing an alphabet which could better fit the Korean tongue and was easier to learn. For most of its history, Koreans have used Chinese for their written language. Written Chinese, Hanja, stems from Korea's cultural, religious, and economic connections with the larger Chinese sphere. Korea has often modeled her government and society to be like "big brother" China. Korean Buddhism was, for a long time, derived from Chinese sources. Korean Confucianism, likewise. And for these reasons, an understanding of Classical Chinese (with notations relevant to Korean speakers) are increasingly relevant the deeper one goes into textual study of these fields in Korea.

In 1444, the Hall of Worthies published a text called Hunminjeongum which launched the new Korean alphabet for the first time to the people. But Classical Confucianism was, by that time in Korea's history, a potent force, and the Confucian bureaucrats found the concept of an easier script not only ridiculous and offensive, but a threat to their economic and political power structure. The Korean script which is so loved and famous by modern Koreans that it has a holiday (October 9) and huge statue in Seoul's literal center, was suppressed in any kind of official capacity until relatively recent times. Even in movies filmed in the 1950s, such as the famous Ja yu bu in have way more Hanja in the streets than you would ever see in economic life in Korea today. There's a great AskHistorians Podcast episode about language policy in East Asia that I highly recommend.

Tibet was a different story, but many of the same forces were at work. Tibetan written language dates to the seventh century, while written Chinese dates much, much older. About a thousand years older than the Tibetan writing system. Tibetan is, actually, a Brahmi script. Probably for more economic reasons (traveling southwards to India is difficult, but quicker than going all the way to China and back in historic Tibet, though there is an exception which I'll come back to), did the Tibetan Empire under Songtsen Gampo decide to adopt a Brahmic script for his Empire, rather than a Chinese one.

While it's pretty clear at this point (to Western Historians, classical Tibetan ones prefer to discuss the advent of Tibetan writing by Thonmi Sambhota as a method to translate Indian Buddhism into Tibetan) that Tibetan script was developed originally for purposes of Imperial Administration, it was a steady stream of Buddhist teachings and translations that sustained the development and proliferation of Tibetan script. During the Later Diffusion of Buddhism in Tibet (from the 11th to 12th Century) Indian pandits helped Tibetan translators write, develop, and diffuse teachings from Indian languages (primarily Sanskrit) into Tibetan. Many of Tibet's medieval clans became wealthy and powerful primarily as well known and prolific translators. The Khon clan, though they trace their lineage to a concubine of the great Tibetan Emperor Tri Song Detsuen, truly got their start as translators of Tibetan Tantras and the development of their own canon tradition. The famous scholar Venerable Buton was the aggregator of the first Tibetan Kanjur and Tanjur, for which he lays out many of his choices and reasons for inclusion in his famous History of the Dharma in India and Tibet, an influential Buddhist history across Tibetan traditions.

The exception that I mentioned earlier is the Tangut (also known as the Xixia in Chinese, and Minyak in Tibetan). The Tangut, who were situated in modern Gansu and Qinghai Provinces were on the Silk Road, centered around Dunhuang, while they spoke a Tibetic language (allegedly) had an economic, religious, and political foot quite deep in the Chinese cultural sphere. Tangut Buddhism is far more related to the Ch'an of China rather than the Vajrayana of India (though it's important to note that the Mongols inherited the tradition of Tibetan Lamas in their court from the Tanguts). Chinese Ch'an tradition had a huge base in Dunhuang, as evidenced by Dunhuang's Cave 17 and the Samye Debate during Tri Song Detsuen's reign between Ch'an supporters of Mo He Yan and Vajrayana supporters of Shantarakshita and his student Kamalashila. The traditional history say that the Vajrayanists won out over the Ch'an (though there is a single Ch'an history that indicates the reverse). And while this is clearly a post-facto edit of the historical narrative, what is clear is that after Buddhism structurally collapsed in central Tibet, Ch'an (bsam gtan in Tibetan) failed to take root in central Tibet, while Vajrayana tradition (see above on the closer economic and cultural contact with India) returned.

But the Tangut, who were closer and on more direct trade routes to the Chinese cultural core, took Chinese Buddhist form as their state religion, used it to justify their Kingship and rule over what seemed quite more like a tribal confederacy more than what we would understand as a monarchy today, and, yes, a Chinese-modeled system of writing.

The Tanguts were destroyed by the Mongols after a poorly decided rebellion in the 1220s. Tibet was conquered in the 1240s and China afterward. For a brief period of Time, Chogyal Phagspa, the hierarch of the Sakya clan (descendant of the Khon translators above) developed a written alphabet we just call "Phagspa Script" today. Originally the intention was to unite the written languages of Tibetan, Chinese, Mongol, and Sanskrit. It was very obviously designed mostly off of Tibetan, but was way too complex and failed to take root anywhere but in ceremonial and ornamental Tibetan Buddhist manuscripts and symbolism.

Phagspa script is usually pointed towards when looking for influences and inspiration in Hangul. The connection is largely speculative, and probably the best connection is expressed as that it was one of the scripts studied by the Hall of Worthies when researching the features that other scripts used in the world.

The Tanguts are gone, but their complex Sinic written language survives in the Tangut Canon. In parts of the Nyingma Canon there are some Chinese texts, though let's not overstate their importance. The Nyingma Gyubum is hundreds of texts long, of which the Chinese texts make up a single volume. Part of this stems from the Nyingma tracing their lineage back to the Imperial period, when Chinese Ch'an tradition was given (at least partly) free reign through Tibet, and Chinese (and Korean) teachers were giving initiations and teachings in Tibet and to the Tsenpos themselves. Some Ch'an tradition remains (allegedly, this is still an understudied field) in modern Tibetan Dzogchen tradition.

Tibetan Buddhism made it to China as a religious rite usually for the wealthy and powerful (more than a few Emperors of the Ming and Qing Dynasty were very devout Tibetan Buddhists) but not as a structural entity where the texts were studied, used, disseminated, and found their way into secular society, the way they were in Tibet, and the way Chinese was never dislodged out of China.

Are Chinese and Tibetan in the same language family? As we say in German, jein. It's unclear what, if any, connection there is. And lots of reputable Tibetologists (independent of the Chinese government) deny a connection. That said, the most reasonable hypothesis for migration into the Tibetan Plateau would be from the Chinese lowlands, so very old linguistic connection doesn't seem out of the realm of possibility.

That said, being in the same language family means very little when determining which symbols can and should express which sound. Hindi, Czech, German, English, and Latin are all Indo-European languages, but only two of them share the exact same alphabet, and we can argue all day over whether English has an appropriate amount of letters or an appropriate orthography (there are four pronunciations of "a" and three for "c," This would be regarded as insanity in Tibetan, Korean, and German).

Tibetan only has two tones (high and low). Chinese has four. Tibetan script has been used, historically, as a translation tool for Indian Buddhism, while Chinese script has historically been a reinforcer of Chinese Buddhism and Confucianism, but even where these ideas spread they had to evolve and change language as well (see Korea and as you mentioned, Japan). Even in the Tibetic case where Chinese was seen as the preferred option (Tangut), changes to the written language had to be made in order to effectively communicate the language and provide justification for its own existence.

So I the question is, why?

Ideographic/logographic scripts such as Chinese have one enormous advantage: they can be multi-lingual. The same script can be used for writing multiple languages in a mutually intelligible way, even though the spoken languages are not mutually intelligible. Given the diversity of Chinese languages, and the lack of an education system that could be used to teach a common language until the Republic, written Chinese performed the function of a lingua franca. This use extended even outside China.

If China had switched to a phonetic script, they would have lost this common written language.

Ideographic/logographic come with a cost: the meanings of the characters must be learned separately from the spoken language. Simplification of teaching the written language (often aimed at increasing literacy rates) was the motivation for, first, the development of the phonetic Korean alphabet, hangul, and second, the South Korean decision to deprecate the use of hanja (Chinese characters) in everyday use and education. (Before this, Korean newspapers were about 50/50 hanja and hangul; hanja remains in use for some official purposes, for example performing the useful function of differentiating between family names that are homophones but are written with different hanja.)

what stopped Japan from switching to using either of its alphabetical writing systems in exclusivity?

... and Korea, too.

Classical Chinese was used as a literary and government language in both Korea and Japan. This meant that those who aspired to be producers of "serious literature" or sought government positions learned to read and write Chinese. Since government positions were highly desired, there was no shortage of candidates literate in Chinese. The system worked, and there was no need to switch writing systems.

There is an ongoing project to make the Joseon Dynasty archives available online. Step 2 is to translate them into Korean, to make them available to Korean scholars and the public, without requiring literacy in Classical Chinese. This is only necessary due to (a) the modern decline in literacy in Classical Chinese in Korea and (b) the desire to make these archives widely available to the public, rather than just to an educated elite. The original purpose was for them to be available to that educated elite, and Classical Chinese served well, including restricting access to the educated elite.

(Early Korean printed literature in hangul was not aimed at the "serious literature market"; it was popular entertainment, often read by women.)