Without PRC China there is likely no stalemate in Korea, if a war in Korea even happens. Even if it does a Manchurian PRC rump state is unlikely to be able to deploy the force necessary to push back UN forces and guard it's border against the RoC who will be looking to finish the job. With the RoC almost certainly becoming a US ally to contain communism in Asia, there is likely no Japanese rearmament. This will reduce feeling in RoC for the need to compete with their old rivals in the naval sphere.
The RoC military will be heavily tilted to building up a massive mechanized land force to protect it's frontier with Soviet aligned Manchuria, and Mongolia, later there will be serious mountain and airmobile forces to defend the western frontier with the USSR. The RoC will also be in range of pretty much all the Russian strategic airforce, which will want as a target since China is big enough that industries in the south will out of range (at least initially) from being threatened by soviet land forces. Thus the RoC will need to also invest in a mdoern interceptor force, along with frontal aviation and tactical air of it's own. Even with heavy US development aid it is not going to have many resources left after building that military and funding industrial development to also develop a large navy.
That said RoC is likely to want some kind of navy for prestige reasons. I would expect a large amount of surplus American ships. Initially this will mean a big batch of old WWII destroyers and some Brooklyn class cruisers. (Probably Savannah and Honolulu since they got scrapped instead of being sold.) They might alter augment this with a British built CVL (likely Leviathan) for a token carrier capability, though I don't see this as being maintained for very long. Most likely dumped in late 70s when the aircraft that fit on them become obsolete, though by this time the Chinese economy might have improved enough to snag some harriers. Leviathan could get a refit to operate them or be swapped for Hermes or a domestically built Invincible type ship. Meanwhile the escort fleet will likely have grown to help the US pacific fleet out with the Soviet Far East navy. More US surplus ships , perhaps combined with some domestically built types. Probably a lot of Perry clones along with a bunch of Spruance type air defence destroyers to replace the old cruisers. The one significant place of investment may be in conventionally powered submarines which would have the mission of hunting down their soviet counterparts in the sea of Japan and protecting Chinese coastal waters. Most of these are probably bought from British and French yards, though later on there might be domestic models too.
Despite maintaining a modest blue water force, the RoC Navywould probably have a significant littoral component of missile boats and fast attack craft with which to contest the Bohai and yellow seas with the PRC'S naval forces.