What the Latin American countries did was ridiculously short-sighted -- they decided that the shift from pre-dreadnoughts to dreadnoughts was their big chance to have naval power on par with the major European powers, because everyone was more or less starting over from scratch. This ignored the fact that the European superiority in fleets was built up from a base of economic superiority; while the Latin powers might grab an early lead by aggressively embracing dreadnoughts, there was literally no possible way they could stay ahead.
I can't speak for the Argentinian and Chilean navies, but what I know of the Brazilian Navy is the following:
- The Brazilian Navy was neglected after the mid-1880s, just as the Argentinians and Chileans were starting their arms race, that went with the race to grab for each of their countries as much of Patagonia as possible; the chaos of the 1890s in Brazil meant that there was no feasible plan for Navy expansion, and whatever was bought was on an emergency basis;
- This means that by 1903 or so(when both political and economic situations were stabilized), the Brazilian Navy was mostly obsolete, quickly approaching the point that it would have to be replaced all at once;
- There were two big questions about what would be the Brazilian Navy of the early 20th century: Should we invest into infrastructure(maintenance and training, mainly) or new ships? Should we buy many, small ships or few, large ones? Investing in infrastructure would be a better base for future growth, but more than half of the fleet needed replacing RIGHT NOW. Smaller ships(coastal defence ships) would cover more ground, but the lack of firepower could be fatal(remember the main force of the Argentinian Navy back then was composed of 4
Garibaldi-class armoured cruisers, a very powerful collection of ships for the time and region);
- By 1905, the decisions were made: modern, large ships would be bought; it was felt that the need for replacement of a large part of the Navy was a bigger problem than the lack of modern infrastructure, and that the large modern ships would be able to give the Navy enough time to invest in infrastructure - in hindsight that would be a mistake. The Brazilian government would acquire 6 13,000 T battleships and 6 7,500-8,500 T armoured cruisers. Argentina, which 3-4 years before had managed to negotiate with Chile an end to their arms race, was already losing its shit over the Brazilian naval expansion, just as the keels for the battleships were being laid;
- Enter Mr. Eustace Tennyson D'Eyncourt, which back then was Vickers representative for South America, and who had ready access to both the Brazilian Navy Ministry and to RN naval plans. D'Eyncourt convinced the Brazilian Navy Minister to cancel the units being built, because there were designs on the pipeline that would make the Brazilian battleships obsolete even before launch. The Navy Minister, Admiral Alexandrino de Alencar, ordered new designs to be presented, something quickly done. As a result, Brazil bought 3 19,000 T dreadnought battleships(the armoured cruisers were put on hold for the moment), and the setting for the South American Dreadnought Arms Race was set.