I can't speak on behalf of South Korea, although there are some interlinking factors between both Taiwan and SK.

In the case of Taiwan, lets rewind the clock back to a few years earlier and do a quick recount of the Chinese Civil War. As you may (or may not) know, the Civil War is considered to have began in 1926 in Shanghai, a Chiang-led betrayal of the unified front which threw the young CCP into disarray and pushed them deep into China's interior. The first rebellions and revolts against the KMT by the CCP were focused in central China, around Hunan, Jiangxi, and Hubei. Here's a political map of China in case you're unfamiliar. After the fall of the Jiangxi Soviet in 1934 most of the CCP leadership fled north into Shaanxi, to Yan'an, and it was there that they would operate for the rest of the period until the end of WW2. After WW2, the CCP found itself in control of a fair chunk of rural northern China, including most of Manchuria which was handed over to them by the USSR after August of 1945. Through all of this, if you follow, you'll notice that the CCP never did quite establish a base anywhere near the coast of China. And even if they had, its not like they had the resources to build up any sort of fleet except for a few junk ships. For the majority of its early existence, the PLA was solely a land-based military.

The year for the establishment for the PLA Navy and Airforce is late 1949, but this is more of a formal creation of branches in order to modernize the nation's military rather than the PLA immediately turning into some sort of military superpower. When the KMT fled to Taiwan, they took all remaining aircraft and ships with them to Taiwan. In fact, the navy played a huge role in funneling troops and officials into Taiwan as they lost the mainland. So at least for the first few years, Chiang knew a naval invasion of Taiwan by the CCP was nearly impossible aside from perhaps the outlaying minor islands such as Quemoy which are miles off the coast of the mainland. Still, 1950 was a year of great concern for Chiang and the KMT. There was a known build-up of ships off the coast of Xiamen and an invasion of Taiwan did seem imminent. Even after the onset of the Korean War, there was no telling what China's ultimate plan was. Militarily, there were only two options: invasion of Taiwan or defense of Korea. At first, it seemed that Korea would fall to the North, but the quick reversal of North Korean gains at Pusan put China on edge. The matter was put to rest later on in that year, when China officially invaded the Korean peninsula in defense of its ally, and found itself entrenched in a back-and-forth war that continued for three more years.

The outbreak of the Korean War was a major turning point in East Asian-American relations. Originally the Americans seemed as though they were prepared to allow CCP forces to cross into Taiwan and finish the job. But the outbreak of hostilities with China convinced America to send the US Seventh Fleet into the Taiwan Straits, more than enough to secure the island against anything that China could mobilize to invade. Throughout Eisenhower's presidency, Taiwanese-American relations reached a sort of zenith, as anti-communist feelings among the American public and politicians climbed steadily. American guarantees to protect Taiwan made sure the CCP would be hesitant in trying to invade the island.

Conclusion

In the late 1950s, the PRC went into a period of economic decline as the Great Leap Forward proved to be a disaster. Going into the 1960s, any military effort to take Taiwan seemed elusive. The 1960-70s were plagued with political troubles in the PRC again as the Cultural Revolution began and devastated Chinese society. By the time Deng Xiaoping took over as chairman, the PRC began to look at more constructive ways to "reclaim" their lost territories, Macau, Hong Kong, and Taiwan, which almost entirely meant economically. Truly, by the early 1950s Chiang knew that retaking the mainland was impossible. And by the 1960s, Mao knew that retaking Taiwan would be pointless. In recent times the two have formed a mutual trading relationship, though that may be disrupted by local Taiwanese dissent against increasing PRC influence on the island.