Outside my area of expertise here, but I would venture that...
This depends on how the conquests proceed and what you believe the effects of Mongol conquest were.
In general, having Japan invaded by another power would change it so that it is less sure of its own isolation. I think it would have more of a precedent of Chinese/Mongolian/Yuan/Korean interference. Perhaps Japan would be more like Vietnam, a country on China's periphery with an exceedingly strained relationship with China, eventually gobbled up in the colonial race before clawing its way back to sovereignty.
Regarding Korea, where the invasion would be launched from, mayhap this experience in conquest and being the main point of contact with Japan would result in a more expansionist society, probably not in the short term, but in event of colonization (I don't see anything here butterflying the Great Divergence), Korea might take Japan's place as the Asian country to build its own empire. (Side note: I think this is one of the few PODs where you could arrange that without multiple PODs, since geography makes it easier for Japan, doing this requires nerfing Japan and making Korea more dynamic in one fell swoop- as this might do).
Regarding the Yuan, I don't know what impact it would have.
This depends on how the conquests proceed and what you believe the effects of Mongol conquest were.
In general, having Japan invaded by another power would change it so that it is less sure of its own isolation. I think it would have more of a precedent of Chinese/Mongolian/Yuan/Korean interference. Perhaps Japan would be more like Vietnam, a country on China's periphery with an exceedingly strained relationship with China, eventually gobbled up in the colonial race before clawing its way back to sovereignty.
Regarding Korea, where the invasion would be launched from, mayhap this experience in conquest and being the main point of contact with Japan would result in a more expansionist society, probably not in the short term, but in event of colonization (I don't see anything here butterflying the Great Divergence), Korea might take Japan's place as the Asian country to build its own empire. (Side note: I think this is one of the few PODs where you could arrange that without multiple PODs, since geography makes it easier for Japan, doing this requires nerfing Japan and making Korea more dynamic in one fell swoop- as this might do).
Regarding the Yuan, I don't know what impact it would have.
This is what the Yuan Dynasty would look like in this timeline.
The other point I will add is that the failed invasions of Japan did not do favors to the Yuan navy. It is therefore possible that the expedition against Java may be butterflied or be more successful.
The other point I will add is that the failed invasions of Japan did not do favors to the Yuan navy. It is therefore possible that the expedition against Java may be butterflied or be more successful.
It's a colossal quagmire that will weigh down the Yuan Dynasty until the invasion is complete, since the logistical stress, constant loss of ships from storms/local Japanese attacks (many provinces had provincial fleets controlled by local clans) will probably delay or cancel the Mongol invasions of Java and likely Pagan/Burma (Pagan will likely collapse irregardless of what the Mongols do). It will also affect the composition of the later Mongol invasions of Vietnam and Champa and make the Vietnamese victories more decisive.
My thought is that we'll have a decisive victory at the landing, followed by the Japanese reorganising in Kyushu for another battle which the Yuan will win resulting in the conquest of Kyushu. Kublai Khan will need to follow this up with an attack on Honshu so will likely send another army to the Chuugoku region. At this point, he will have at least a few turncoat clans on his side. If the Yuan win in Chuugoku and gain a foothold on Honshu, they'll set off a chain of events that leads to the end of the Hojo regents and the entire Kamakura Shogunate because there will be plenty of clans which feel the government can no longer protect them and for the ones who don't go over to the Mongols they will form their own associations that will likely dethrone the Shogunate.
The struggles of the Nanboku-cho era between the Northern and Southern Courts based on a succession dispute date from the era immediately prior to the Mongol Invasions, so there is plenty of reason to think the Mongols could find a suitable puppet "emperor". I doubt he will be allowed to maintain that title so he might receive a different title befitting his religious position or might simply rule as shogun and "king of Japan".
Naturally, the Japanese will not immediately lose once the Mongols conquer Chuugoku and sack Kyoto, so likely the Japanese will regroup in Eastern Japan closer to Kamakura. Kublai Khan likely will have died before then and the various issues within the Yuan Dynasty will start cropping up. If the Yuan and their Japanese allies invade again, they will very likely be defeated by the Eastern Japanese clans (probably around Sekigahara as that's the natural route between Eastern and Western Japan). The leaders of the army will likely include the ruler who supplants the Hojo regents and will be able to do whatever he wants, be it set up his own shogunate or even serve the Emperor (Emperor Go-Nijo, Go-Daigo's older brother) directly and restore him to power in an early Kenmu Restoration. IMO, this is by far the most likely result of a successful Mongol Invasion of Japan--a divided Japan.
Theoretically, the Mongols can win of course and conquer Kanto and Tohoku. But the amount of manpower and ships expended will probably lead to an even uglier collapse of the Yuan Dynasty. Japan will be ruled by a local vassal. Incidentally, I like the potential of this guy, Mori Tsunemitsu (ancestor of the famous Mori Motonari) playing a prominent role in Mongol Japan, given how his father ended up on the wrong side of the Hojo clan (and was killed for it). Tsunemitsu should be in his 40s-50s during the Mongol Invasions so he could survive into the 1290s-1300s and become one of Mongol Japan's most prominent local generals.
Mongol collapse in Japan is only a matter of time and probably will occur concurrent to the Red Turban Rebellion in China. At that point, it's all up in the air what happens in Japan. My guess is an imperial restoration similar to the Kenmu Restoration and potentially an even more violent Nanboku-cho era aimed at expelling Korean, Chinese, and other influences from Japan (basically sonno joi centuries early). Those non-Japanese who settled in Japan will likely be subject to massacres, like the Muslims (foreign or local converts) who will have settled there (as they did in Korea).
What happens after I'm not sure. It could be a much more open, naval-focused Japan with a sense of unity having basically completed a "reconquista" that's ever-wary of Korea and Japan. Japan could go right back to fights between powerful clans. Maybe they colonise Ezo/Hokkaido early (after all, the Mongols and their Nivkh allies did fight the Ainu OTL, so maybe Hokkaido is seen as a security risk to not colonise and a chance to give good clans some extra land), and maybe this spurs the classic "Japan colonises America" scenario, or at the very least makes a "Japan as Britain vs China as France" scenario. The potentials are pretty endless.
My thought is that we'll have a decisive victory at the landing, followed by the Japanese reorganising in Kyushu for another battle which the Yuan will win resulting in the conquest of Kyushu. Kublai Khan will need to follow this up with an attack on Honshu so will likely send another army to the Chuugoku region. At this point, he will have at least a few turncoat clans on his side. If the Yuan win in Chuugoku and gain a foothold on Honshu, they'll set off a chain of events that leads to the end of the Hojo regents and the entire Kamakura Shogunate because there will be plenty of clans which feel the government can no longer protect them and for the ones who don't go over to the Mongols they will form their own associations that will likely dethrone the Shogunate.
The struggles of the Nanboku-cho era between the Northern and Southern Courts based on a succession dispute date from the era immediately prior to the Mongol Invasions, so there is plenty of reason to think the Mongols could find a suitable puppet "emperor". I doubt he will be allowed to maintain that title so he might receive a different title befitting his religious position or might simply rule as shogun and "king of Japan".
Naturally, the Japanese will not immediately lose once the Mongols conquer Chuugoku and sack Kyoto, so likely the Japanese will regroup in Eastern Japan closer to Kamakura. Kublai Khan likely will have died before then and the various issues within the Yuan Dynasty will start cropping up. If the Yuan and their Japanese allies invade again, they will very likely be defeated by the Eastern Japanese clans (probably around Sekigahara as that's the natural route between Eastern and Western Japan). The leaders of the army will likely include the ruler who supplants the Hojo regents and will be able to do whatever he wants, be it set up his own shogunate or even serve the Emperor (Emperor Go-Nijo, Go-Daigo's older brother) directly and restore him to power in an early Kenmu Restoration. IMO, this is by far the most likely result of a successful Mongol Invasion of Japan--a divided Japan.
Theoretically, the Mongols can win of course and conquer Kanto and Tohoku. But the amount of manpower and ships expended will probably lead to an even uglier collapse of the Yuan Dynasty. Japan will be ruled by a local vassal. Incidentally, I like the potential of this guy, Mori Tsunemitsu (ancestor of the famous Mori Motonari) playing a prominent role in Mongol Japan, given how his father ended up on the wrong side of the Hojo clan (and was killed for it). Tsunemitsu should be in his 40s-50s during the Mongol Invasions so he could survive into the 1290s-1300s and become one of Mongol Japan's most prominent local generals.
Mongol collapse in Japan is only a matter of time and probably will occur concurrent to the Red Turban Rebellion in China. At that point, it's all up in the air what happens in Japan. My guess is an imperial restoration similar to the Kenmu Restoration and potentially an even more violent Nanboku-cho era aimed at expelling Korean, Chinese, and other influences from Japan (basically sonno joi centuries early). Those non-Japanese who settled in Japan will likely be subject to massacres, like the Muslims (foreign or local converts) who will have settled there (as they did in Korea).
What happens after I'm not sure. It could be a much more open, naval-focused Japan with a sense of unity having basically completed a "reconquista" that's ever-wary of Korea and Japan. Japan could go right back to fights between powerful clans. Maybe they colonise Ezo/Hokkaido early (after all, the Mongols and their Nivkh allies did fight the Ainu OTL, so maybe Hokkaido is seen as a security risk to not colonise and a chance to give good clans some extra land), and maybe this spurs the classic "Japan colonises America" scenario, or at the very least makes a "Japan as Britain vs China as France" scenario. The potentials are pretty endless.
I think the post-Yuan situation would result in Japan violently expelling the Koreans, meaning Japan will have a navy and a sizable army and likely be less divided (since whoever expelled the Mongols will end up as shogun).