The US had already acquired much of the world's gold reserves by the end of World War II. This was due primarily to the way that gold was used back then as a medium of exchange.

In the current monetary system, currencies are typically fluctuating based off confidence in a country's financial system. Investors and savers looking to protect their portfolios will flock to the most stable currency with the idea that the value of said currency will change little (less inflationary pressure from high growth or government spending, for example). Similarly, in addition to speculating in currency, they may also choose to invest into precious metals, gold being one of the main ones for its relative stability.

This was not the case of the monetary system in the early 20th century. In the early 20th century, most currencies were backed by, if not actually made out of, gold or silver. The former type of gold, where notes were issued that could be exchangeable (or to be more precise, convertible) for gold, is known as gold bullion (i.e. gold bars). The latter, where the currency is made out of gold (as a coin), is known as gold specie.

Now as a (poor) thought exercise, imagine that you are a factory owner in the United States. You buy, say, dye and wool from a British company, and use it to make clothing that you then sell to a company in Britain. If the dye and wool costs you $5 and you sell the clothing for $10, in this transaction the net difference, $5, is the amount you would earn, presumably in hard cash.

Across many such transactions, it would seem that the US is exporting to Britain much more than it is importing. Britain needs to pay for this $5 per transaction somehow. Since they are not paying it through the transfer of goods, they need to essentially give me US dollars, instead.

How do I get a US dollar in Britain? It's fairly straightforward. I would take my British pounds and exchange it for gold. Then, I would exchange the gold to a US bank to get US dollars. Finally, I would use the US dollars to pay off the existing balance.

The key here is the phrase "exchange the gold to a US bank." Because gold was valued by all countries and had a defined exchange rate to the local currency, it naturally became a means of international trade, allowing one to buy and sell currency as needed to buy and sell real goods.

Now, what happened in the early 20th century was an event known as World War I. This event neatly destroyed the careful financial system. Why?

Due to the wartime situation, the governments of each country needed to spend money to buy goods, including weapons, raw materials, food, silk for uniforms, etc. Because of the large manpower demand of the militaries, this resulted in a labor shortfall in the countries at war, which meant that they needed to find external sources of these goods.

Lo and behold, the United States was for the first few years of the war blissfully neutral. In the case of Britain and France, they eagerly sought the US as a trade partner to acquire these goods. The problem was that the war dragged on, and soon both Britain and France ran out of gold to exchange for US dollars and therefore US goods. The US graciously extended the countries loans (of gold, to buy US dollars-in practice they skipped the middleman, but the British and French were still on the hook for the gold).

In the aftermath of the war, the United States, being one of the few countries that had not been devastated by the war (Britain, while relatively unscathed, had a massive burden of loans that they needed to pay off) became a natural trade partner for many countries, who, while war industries retooled for civilian use, soldiers returned to the workforce, and so on, ended up forking over a good amount of gold in exchange for US products, be it consumables or capital expenditures like machine parts. Many countries were concerned about staving off inflation, as the massive government spending that went into the war with a corresponding decrease in real goods resulted in a potentially inflationary situation that needed to be dealt with. A notable exception was Weimar Germany, who, in an act to attempt to throw off the "shackles" of reparations, chose to destroy their economy.

The reparations of Versailles required the Germans to pay reparations in the form of either real product (such as coal) or in the form of gold (that could be used by Britain and France to rebuild their reserves, or rather, to pay off the US). The reparations themselves were cleverly designed to be relatively minor in impact while appearing to be much larger in scale to salve public opinion in Britain and France. However, for Germany, there was strong anger against the Treaty of Versailles in general, the reparations being a notable target, and thus the government set out to find ways to undermine the treaty and the reparations. To do so, the German government, which had abandoned the gold standard in 1914 to be able to print money for deficit spending, chose to buy gold in exchange for the Papiermark, the paper mark currency used since 1914. They deliberately printed Papiermarks to continue buying gold, which led to hyperinflation in Germany, known as the Weimar Hyperinflation (which to this day continues to be confused with the deflationary aspects of the Great Depression). This in turn did lead to a renegotiation of the reparations, but only at the cost of massive wealth destruction in the German economy. This is as discussed in Sally Marks' Myth of Reparations.

For simplicity's sake (and due to the massive ongoing academic debate between economists about the causes and solutions to the Great Depression) we are going to bypass it entirely for purposes of this discussion, I will mention that some theories are that the Federal Reserve did not act to increase money supply, another that the gold standard itself prevented the Federal Reserve from doing so and that the economic power of the US made such paralysis global, among many others (ex. Keynes' consumption based theory, and so forth). I will go on to mention however that there was major slowdown in international trade (and thus, currency exchange) and some countries like Britain abandoned the gold standard in order to deficit spend in an attempt to alleviate the worst effects of the depression. This, combined with the outbreak of World War II (where, for one example, through the "cash and carry" policy, Britain exchanged much of its liquid cash reserves (incl. gold) for US goods) resulted in the US being the dominant player in gold reserves by 1944.

Sources:

Friedman, A Monetary History of the United States

Eichengreen, Golden Fetters: The Gold Standard and the Great Depression

Follow up question: I thought South Africa was a gold-rich country. What happened to all of it?