Argentina, Chile and Brazil did not need a strong navy, especially not multiple dreadnoughts. They didn't have any overseas possessions nor any strategic necessities for one. It is kinda like Germany before WW1, except that Germany at least had overseas possesions to protect.
So why then did it happen, how did it happen and how was it financed?
While the dreadnought race was the largest and most expensive, there were two naval arms races in South America during the late 1800s and early 1900s. The first would pit Chile against Argentina, while the second would add Brazil to the mix. The main driver of tension between Chile and Argentina were several border disputes, as well as the ownership of the disputed Beagle Channel. There was little tension between either Chile or Argentina and Brazil, but the Brazilian naval build-up threatened to overturn the balance of power.
The first naval race was sparked, in 1887, by the Chilean ordering of a new ironclad battleship, the Capitan Prat. This was accompanied by orders for two new cruisers, two torpedo boats, and weapons for the expansion of the Chilean Army. This seriously threatened Argentina - with the new ships, the Chilean navy would be more experienced and have bigger ships, enough to counteract Argentina's advantage in numbers. The Argentinian navy responded by ordering two more battleships, and two new cruisers. In 1891, Chile underwent a civil war, which failed to loosen tensions. The Chilean Navy had been a major supporter of the winning side, and so its star was in the ascendance. This led to an increase in spending, most of which went on buying new cruisers. The Argentinians responded in kind, driving tensions higher. In 1898, the Argentinians ordered two Italian armoured cruisers, to which Chile did not have an answer. This seemed to the Chileans like a prelude to war, so they moved towards arbitration of several results, calling in the British. However, the arbitration took time, and the naval race continued. The Chileans ordered two new fast battleships in 1901, to which the Argentinians responded for by publishing tenders for a new battleship. This drove tensions further, endangering British trade interests in both countries. The British stepped in, and insisted on a negotiated solution to the crisis. Both countries were also beginning to suffer economically, which helped to drive them to the negotiating table. Argentina was struggling to integrate a large amount of new immigrants, while Chile's smaller economy was suffering under the pressure of the arms race. Talks were held in Santiago, Chile's capital, which led to the signing of the 'Pacts of May' in May 1902. These assigned territories and spheres of influence between the two countries, arbitrated by the British. More importantly for the naval side, they included an arms limitation pact. The two Chilean battleships were sold to the Royal Navy, becoming the Swiftsure class, while the two Argentinian cruisers became the Japanese Kasuga class. Several older ships were disarmed, and a five year moratorium placed on new warship acquisitions.
This pact lasted until 1906, when Brazil placed orders for new battleships. This was the result of a 1904 decision by the Brazilian Congress to acquire new warships, funded by a boom in rubber and coffee. The Brazilian Navy had also been the main regional power before the 1889 revolution, and they hoped to recover this standing. Following a year of indecision as to how the new fleet should be composed, two new dreadnoughts were ordered from British yards (with a third to come once they were delivered). The order was seen as suspicious by some, as this order would result in Brazil getting dreadnoughts before more established naval powers like France and Russia - rumours quickly spread that Brazil was acting as an agent for some third party. This plan was a direct threat to Argentina, with its building constrained by the Pacts of May. The response decided upon was to cancel the treaty, and procure new ships. However, there was fierce internal dispute about this, as many saw the Brazilian plan as unaffordable, while the hardliners worried about a pre-emptive strike. There were even suggestions that Argentina should attack Brazil before the ships could be delivered, should Brazil refuse to cancel them (though the Argentine population did not support such decisive action). In August 1908, an Armaments Procurement Act was passed, which called for the spending of $55 million on new warships. This led to intense competition for the contracts, with no fewer than 144 tenders being submitted by firms from five countries. In 1910, orders were placed with the American Fore River Company for two new 30,000 tonne battleships. This was a first for American shipbuilding, which had traditionally lagged behind its European competitors. It was also a win for the State Department, which saw it as a way to encourage solidarity between nations in the Americas. The Chileans were late arrivers to the race. While the resumption of building in Argentina was a threat to Chile, the Chilean economy was not initially able to support new naval building. The nitrate market (one of Chile's main exports) had crashed in 1907, and then in 1908 an earthquake had devastated much of the country. However, a brief war scare with Peru helped push Chile towards buying new ships. Thanks to close ties to the RN, the Chilean Navy placed orders for two new battleships with the British company of Armstrong Whitworth.
However, shortly after the Chilean orders, the naval race began to slow down. The two Brazilian battleships had been delivered in January and July 1910, but in November of that year, the Brazilian Navy experienced a severe mutiny. Nicknamed 'the Revolt of the Lash', sailors seized several ships, including the two battleships in protest against poor treatment and conditions, including excessive use of corporal punishment. To bring their point home, they bombarded Rio de Janeiro, while military action by loyalist ships was hamstrung by the Brazilian Admiralty's unwillingness to fire upon their most expensive and prestigious ships. Negotiations with the mutineers were ultimately successful, leading to the end of corporal punishment and an amnesty for all involved. The mutiny led to a lack of confidence in the navy, and many ships, including the two dreadnoughts, were effectively disarmed to prevent any re-occurrence. Additionally, Brazil's economic situation was beginning to turn, as British rubber plantations opened in the Far East and a European depression caused loans to dry up. While a third (excessively-armed) battleship had been ordered, she was sold to the Ottoman Empire in 1913. This reduced tensions, allowing Argentina and Chile, which had also been suffering monetary troubles, to consider selling their ships. There was considerable popular demand for the money so raised to be spent on internal improvements - welfare programs and nationalised services - rather than on the fleet, feelings echoed in Brazil. The Brazilians did place tenders with Armstrong Whitworth and Vickers for new battleship designs in 1914. However, the beginning of WWI would effectively draw a close on the South American dreadnought race. The two Chilean battleships building in the UK were requisitioned by the RN, with one becoming the battleship Canada, and the other being converted into the carrier Eagle (while the Brazilian battleship sold to the Ottomans became the British Agincourt). The shipbuilders of the belligerent nations closed their doors to foreign customers - and as none of the South American countries had sufficient industry to build such ships themselves, they were dependent on the foreign yards that were no longer available. There were some calls to reopen the naval race following WWI. In December 1920, the Brazilian Congress was presented with a bill to purchase new warships, but rejected it due to the poor state of the Brazilian economy. Canada was returned to Chile, becoming the Almirante Latorre, in April 1920. There were calls for the purchase of additional ships to replace Eagle. One possibility for this was the sale of one or both of the remaining Invincible class battlecruisers, but the British price, though low, was seen as still too high. The dreadnought race never brought the three countries close to war, unlike the earlier race between Argentina and Chile, mainly because of internal calls for domestic spending which were less prominent in the earlier race.
Sources:
The Role of the Royal Navy in South America, 1920-1970, Jon Wise, Bloomsbury, 2015
The British Battleship 1906-1940, Norman Friedman, Seaforth, 2015
Warships for Export: Armstrong Warships 1865-1927, Peter Brook, World Ship Society, 1999
Jane's Battleships of the 20th Century, Bernard Ireland, Collins-Janes, 1996
Battleship Diplomacy in South America: 1905-1925, Seward W. Livermore, The Journal of Modern History 16, no. 1, 1944