To answer this question in some detail I have to go a bit further back in time, because I don't want to asume the extent of your knowledge, or of any other person who read this.

At the time, the territory now considered Uruguay, and a big portion of what today is Rio Grande do Sul, was called "La Banda Oriental" (because it was a strip of land east to the Uruguay River). The first revolutionary wars in this territory were leaded by José Artigas, who advocated for a Federal government as a member of a hypothetical nation called "Provincias Unidas del Río de la Plata (United Provinces of the Río de la Plata). The government of Buenos Aires was against this proposal, and supported the idea of a centralized government of the Provinces. This conflict between Unionists and Federalists would have a great impact in the history of the region.

A brief chronology:

In 1811 Artigas defeated the Spanish royalists armies.

In 1813 Buenos Aires refused to acknowledge Banda Oriental's delegates because they tried to push economic and political independence for the Provinces. Artigas broke relations with Buenos Aires and sieged Montevideo.

In 1815 Artigas had militar and political control of the Banda Oriental, and founded the Provincias Unidas del Río de la Plata (six provinces) under his protection. Yet, in the following years Buenos Aires outmanouvered him politically, and undermined his support.

In 1816 the Portuguese Empire invaded the Banda Oriental. Artigas ask for help to Buenos Aires, but he refuses to pay the price for their support (abandon his federalists claims and become a part of a centralized government). Artigas make a camp in the North, and struggle against the Portuguese, who control almost the whole territory including Montevideo.

In 1820, the Banda Oriental is already part of the Portuguese, and later the Brazilian Empire, and is known as Provinicia Cisplatina. Artigas is defeated and exiled to Paraguay. Due to the position of the port of Montevideo relative to the port of Buenos Aires, the Brazilians now have a strong control over the trading in the Río de la Plata.

Due to this, Brazil starts exporting leather in huge quantities. In this years, leather had a huge demand in Europe, not only for manufacturing boots and other clothes, but for the machines of the industrializing British economy. Also, a lot of grain went to Europe from the Río de la Plata commerce. So not only the government of Buenos Aires, but also cattle ranchers from Argentina had a high interest in retaking Montevideo.

In 1825, they financed an expeditionary party later known as "Los Treinta y Tres Orientales" (thirty three orientals). A small group of men from Banda Oriental and other Provinces (more than thirty three though), who crossed the River Uruguay in the night, avoiding brazilian patrolls. Lavalleja was the leader. They brought arms and money to finance a revolution against the brazilians, and had several military success, severely hindering Brazilian presence in the territory. In 1825, independence of the Banda Oriental from Brazil was declared. So, the whole affair of independence was really a Buenos Aires militar and political move to regain control of the Río de la Plata.

The Indepencence Declaratory started a long war (500 days war it's known as) between Brazil and Argentina for the control of this territory, where none could gain the upper hand. It's said that Brazil won the maritime war, and Argentina the land war. This conflict deeply affected the commerce between Europe and the Río de la Plata, so the British crown sent an emisary with the express mission of cease the hostilities between this two powers. So, there came Lord Ponsonby. After several unsuccessful negotiations between Argentina, Brazil and the British, where Orientals (as they were known) had almost no participation, Lord Ponsonby had to resort to the last measure in his plan, that was creating a buffer state between the two powers. This solution was equally unsatisfactory for both Buenos Aires and Rio de Janeiro governments, so it was accepted in 1828. No Uruguayan was present in that meeting. Lord Ponsonby later did the same with Belgium.

So that was the birth of the República Oriental del Uruguay (is more an adress than a name: The Republic that is East of the Uruguay River).

In the treaty signed by the British, Brazilians and Argentinians there was a clause that allowed them to intervene in the new country if there was significant trouble. So if there was political or civil unrest, bith Brazil and Argentina could rightfully invade the country. In 1830 the Constitution was sweared, and Lavalleja was the first president. In the next century, Uruguay would experiment a bloody period of civil war, with severalarmed revolts against several governments, the last in 1904.

Both Argentina and Brazil took part in this conflicts, that affected not only the new nation but the wohle region. Inside Uruguay, there were political factions that promoted the anexation of the country to Argentina, and to Brazil. The main reason this never happened, and neither power used the civil wars to claim the right of invasion to "settle things" was that doing so would justify the invasion of the other power as well, breaking the balance and re-starting a costly war for both sides. They chose economical and political dominance, instead of direct invasion, supporting one or the other side of the numerous revolutions and civil wars fought in Uruguay during the XIX century.

For example, Brazil gained the whole northern half of the Uruguayan territory as a payment for it's intervention in favour of the government in the revolution of 1864, and later forced Uruguay to participate in the War of the Triple Alliance.

So, in short, the Uruguayan Independence from Brazilian Empire was financed and supported by Argentina, with the objective of annexing the Banda Oriental. After the 1828 Montevideo's treaty, there were no serious attempts neither from Brazil or Argentina to formally annex Uruguay, but there minor were political groups that advocated for doing so. Yet, both governments knew that an open war was going to be unpopular, costly, and with low success probabilities, so they prefered meddling in a more indirect form in Uruguay national politics.

Fun fact: Lord Ponsonby was almost killed by a mob during the French Revolution, but was saved by some women because he was "pretty". Our existence as a country is owed to them.

Sources: I don't know any source in English (I am sure there are), but in this link there a serie of radio columns by a history professor that tells the Uruguayan history until the XX century. The XX century is still in production. They are in Spanish of course. If anyone wants to know more about Uruguayan history, feel free to comment or PM. I am not an expert (my degree is in biology), but I have read and researched a lot about this topic because I am writing novels about Uruguayan history in the XIX century, so I have a fair idea where to go looking for particular answers.

Edit: For some reason, I had written 19xx in almost every date from the XIX century. Edited to 18xx thanks to /u/Soft-Dark-3454 comment.