Largely guesswork, didn't bother projecting past 2040, maybe a little optimistic:
Japan - late 2020s
South Korea - 2030s
China - mid-2020s
South Korea is by far the most socially liberal of these three, and why would PRC legalize it?
PRC is by far the most socially liberal of these three when it comes to this specific type of issue. PRC has the least internal opposition, both from the public and from institutions. And Chinese culture is much more liberal than Korean or Japanese culture generally. I don't think the PRC government is particularly interested in social liberalization, but I can also see them enacting same-sex marriage because it seems reasonably popular with the public and there is international interest in them doing so, i.e., they don't really care about the issue but would also move quickly if they thought doing so was advantageous. I do think legalization in Taiwan puts some pressure on the PRC as well.
South Korea is not socially liberal at all, and evangelical Christianity has far more influence in South Korea than in either of the other two. I agree that Moon Jae-in himself probably is personally in favor of legalizing same-sex marriage now, but even within his party he would face stiff resistance, and he's not talking about it at all because there would be significant backlash (more than Tsai Ing-wen faced in Taiwan, and from a particularly religious Christian population especially). Japan would mostly be slow due to apathy/the slow grind of accomplishing any social change in Japan rather than particular principled opposition.
