The People's Republic of China includes mainland China, Macau and Hong Kong. The last two are Special Administrative Regions of the PRC. According to the PRC, it also includes Taiwan.
On the other hand, Taiwan is occupied by the Republic of China, which claims all territories of the PRC.
They both recognize themselves as the only legitimate governments of all China (mainland, Macau, Hong Kong, Taiwan and surrounding islands). So obviously, the ROC would make a claim.
They wouldn't be the only ones to make a claim by the way. There are a lot of natural resources and technology (including WMDs) to loot.
You might think the ROC has a stronger claim than other countries, however Taiwan isn't a member state of the UNSC, and that's a big deal.
A bit of history: after WWII, the United Nations was created, with a United Nations Security Council. 5 states were chosen as permanent members of the UNSC. These states hold the power of veto, which means they can say no to pretty much any decision of the UNSC unilaterally. That is a lot of power.
The winners of WWII were the USA, the UK, France, the USSR and the Republic of China. Logically, they were the first Permanent 5. When USSR fell, the spot was transferred to the Russian Federation, considered as the rightful successor state to the USSR.
Before that, in the 70s, the ROC spot was transferred to the PRC. The PRC became the legitimate representative of China, and the ROC was expelled from the UNSC. The Republic of China is currently unrecognized by the UNSC, and will never be as long as the PRC has the right to veto.
It's possible that Taiwan would become recognized again, but the Chinese spot as permanent member would likely either remain empty or be given to another country. Taiwan would have to beat Germany, India, Brazil and Japan who already want a spot. They have more political, military, cultural and diplomatic power, which makes all of them better candidates than Taiwan.
To answer the question, the situation after one month would be unresolved. It could take year before the UNSC figures out how to reform. Then it would take even more time to decide what to do with China. A lot of countries would be interest in that territory, if even one of them has veto power (like Russia for instance), it's very possible the situation would never get resolved.
Trying to get control by force would be illegal and either fail or start something nobody wants. Nations with claims over the territory would likely do everything they can to preserve the status quo rather than having another nation than them taking control. That could be an interesting scenario for a WWIII though.
Anyways, I can see a few ways this ends.
Option 1: all the countries with claims find an agreement, said agreement is approved by the UNSC, and China is split in many parts going to all interested parties. Obviously, that will take some time.
Option 2: individual people migrate to China and form a new independent country, or several of them. The situation would eventually resolve itself, but that would also take a long time
Option 3: no resolution. China permanently becomes an international zone under UN mandate. Since nobody can agree, nobody gets it.
Option 4: World War III. Everybody tries to rush in illegally. Since they'll inevitably run into each other, things will escalate. With Russia, India and Pakistan having nuclear power, it could go south very quickly.
Option 2+3: A new country is formed under UN administration. People are allowed to immigrate. Once there's enough people to make a country and a government, the UN mandate ends and we have one or several new countries.