I briefly skimmed the English and Korean Wikipedia articles, and it looks like it would be essentially impossible for a successful uprising.
The tension between the citizens and the police occurred because of issues such as high taxes, famine, and infectious diseases such as cholera. The first clash occurred a year before the main uprising (the English version somehow combined the two), in which a child was trampled to death by a horse mounted by police. This caused the locals to throw stones at the latter, which fired back because they didn't understand the reason for the negative sentiment.
In April 5th, 1948, shortly before the general elections in preparation for the establishment of the ROK in August, about 250 locals, who were part of the Workers' Party in Jeju, were afraid that they would lose support after the election, and coordinated attacks on several police stations. However, they failed to realize that the majority of locals opposed the communists, and that the US would most likely quash any event linked to communism. In response, the government sent Kim Ik-ryul, a general, in order to to negotiate with the leader of the rebellion, Kim Dal-sam, and discuss issues with the locals. However, the efforts fell apart as neither side was willing to concede, and the general was wrongfully branded as a communist, causing him to return to the peninsula. Meanwhile, the situation continued to deteriorate, as people's sentiment continued to flare up, causing voting to be erratic as several stations and candidates were attacked.
The continuing instability, along with the president's view that the chaos was a challenge to his authority, finally caused the ROK to send troops in conjunction with the US in order to brutally suppress the locals, although most of the atrocities were carried out by the Korean forces. As a result, most resistance was quashed by the end of the year, although occasional revolts flared up until 1954 mostly due to the Korean War. In the process, about 95% of the villages were burned down, regardless of whether the individuals were communists or not, and many were forced to flee to Japan in order to escape the chaos.
Looking at the whole picture, Jeju was geographically too far from North Korea to receive any substantial support, while the US would have more directly intervened if the uprising had been more successful. The fact that during the war, anyone suspected of being communists were deported to Jeju and executed, suggests that the island was firmly under ROK control by 1950. In addition, the locals were too few in number to resist a sizable force, whether Korean or American, so unless the North ended up unifying the entire peninsula, it would be almost impossible for the uprising to ultimately succeed.
I imagine even if forced of the peninsula, the island would be held by the UN as sort of a Taiwan of Korea for its strategic importance.
Well, no. Taiwan still retained substantial infrastructure that the Japanese had installed during its occupation, while Jeju had virtually none. The US might be able to patrol the island in order to prevent it falling into communist hands, similar to what it did with Taiwan, but that wouldn't resolve the issue of maintaining a stable government and creating a developing economy. Jeju itself would probably remain extremely impoverished unless there were significant exoduses, in which numerous skilled workers managed to increase the population to about 500,000 by 1970 or so. The government itself would also be extremely dictatorial in fear of the peninsular state to the north, and would almost certainly end up conducting widespread purges and oppressing people's rights, which would be much worse in a state with a much smaller area and population.
As a result, if the DPRK somehow managed to take over the peninsula, which would be almost impossible after the US entered the war, there would probably be two options. The US would probably either withdraw from Korean territory altogether after losing Busan, causing the North to take over the island, similar to what occurred in Hainan during the Chinese Civil War, or would eventually give up the island as a liability after decades of failure in Jeju itself.
