Acts such as signing a treaty are done under the auspices of the treaty "depositary", in this case the Secretary-General of the United Nations (UNCLOS Article 319). The depositary's functions in international law are essentially administrative or clerical in nature. They receive and process communications from states, but don't deal with any significant points of disagreement. That is ultimately up to the nations themselves, perhaps through the medium of institutions such as the International Court of Justice.
If you are obviously not a real country, then the depositary is under no obligation to take your communications seriously.
There are many scenarios for genuine disagreement, such as the name of a country even when everybody acknowledges the country is real; the extent of its control over disputed territories; which of its supposed officials are actually empowered to act; etc. Generally, the diplomatic solution is for the depositary to pass the buck to the signatories in thrashing out any actual legal or political consequences.
For multilateral treaties, the fact that one state has signed does not mean that other states are bound to recognize it for all purposes. Sometimes, the other countries will say "we don't think that the Republic of Bob is a real country but we acknowledge that they claim to have signed the treaty". An example is that the USA routinely did this for treaties signed by East Germany. Even in the absence of such a declaration, the understood rule is that no automatic recognition follows. A depositary state is also able to process ratifications from countries that it itself does not recognize.
Therefore, even if the Secretary-General were to add your country to the list of UNCLOS signatories, the other countries are not forced to start treating you as a peer.
In more detail...
For UNCLOS, accession is possible by "States" and certain other entities (Article 307), who can deposit their "instruments of accession" with the Secretary-General. Such an instrument is a formal diplomatic document; the UN imposes some technical requirements on what it has to say, and on whose authority it can be presented (see its Treaty Handbook at 3.3.4 and Annex 5). The depositary will keep track of the overall status of signatures and ratifications, and also forward the information to other parties as changes arise. For this Convention, there is no vote of existing signatories or anything like that; other cases like the EU treaties may have their own different rules. It's also fairly common in drafting to specify that a treaty is open to UN Member States only, or a similar restriction.
Under the Vienna Convention on the Law of Treaties (1969), Article 76(2),
The functions of the depositary of a treaty are international in character and the depositary is under an obligation to act impartially in their performance.
Article 77 further says that the depositary can check that submitted instruments are "in due and proper form", and must bring certain disputes to the attention of the existing signatories. The effect is that the depositary is not meant to bring their own political opinions into play and it's up to the treaty parties to decide what to do about anything of more than administrative significance.
Customary international law allows the Secretary-General (or indeed any depositary) to simply ignore an "accession" by some entity that nobody else thinks is a real State. This is similar to the practice in the Treaty Handbook where a "reservation" - a declaration by the signing state giving details of what it considers its obligations to be - can be refused if it is obviously contrary to the purpose of the treaty.
Such was the situation in 1894 when James Harden-Hickey tried to join the Universal Postal Union in his capacity as self-proclaimed Prince of Trinidad (an island off the east of Brazil, not the one in the West Indies). The Swiss Federal Council was the depositary for the UPU treaty but declined to accept his application until he had achieved recognition from the great powers of the day. See Revue générale de droit international publique, vol. 1 (1894), p179.
There is not a precise normative threshold that we can identify, given the complexity of possible state relations, and the fairly limited number of actual examples. A government-in-exile has generally been able to perform treaty functions, if a majority of the other signatories accept it (Stefan Talmon, Recognition of Governments in International Law, Clarendon Press (1998) at p137). But again, what truly matters is not your country's name going on the list, but the actual reality of how other countries deal with you when it comes to the legal and political consequences of the treaty.
