I can only give a partial answer, that doesn't address at all the question of broader flows of bullion over time. However, I can confirm that a) New World gold and silver ('including Inca and Aztec treasure from the time of the conquistadors') was part of the reserves held by the Bank of Spain as late as the 1930s and that b) the Spanish Civil War of 1936-9 saw the vast majority of Spain's gold reserves used to fund the war effort, leaving little, if any, left in the Spanish treasury after the end of the conflict.

Point A is uncontroversial, though I couldn't give you an exact breakdown of what proportion of bullion reserves came from where or were in what form. Point B, however, is one of the enduring controversies of the civil war, and more particularly its international context - namely, whether Soviet intervention was motivated by anti-fascist altruism, or was a cynical ploy to cheat Spain out of its gold reserves by overcharging for weapons and other aid. The current historical consensus, I think, is somewhere in the middle - that the USSR was motivated to intervene for ideological and strategic reasons, but the aid provided was on a relatively straightforwardly commercial basis.

In any case, on the eve of war in 1936, Spain was home to about $700,000,000 worth of gold (in 1936 prices), excluding the artistic or historic value of individual pieces or coins, held at the Bank of Spain in Madrid. In maintaining control over Madrid on the outbreak of the civil war in mid-July 1936, the Republican government therefore maintained control over Spain's gold reserves. They remained in Madrid until September 1936, when the decision was made to move the gold to Cartagena as Madrid's position grew more precarious in the face of Nationalist advances (and, apparently, rumours that anarchist factions were considering raiding the bank). Cartagena, far from the front and the main base of the Republican Navy, was considered the safest place for it.

Even before its transfer, Spain's gold reserves were being used to fund the Republican war effort. Literally within days of the military uprising (authorisation came on July 24th), gold was dispatched from Madrid to Paris to buy French weapons. When shortly afterwards on August 8th, France decided to adhere to the new Non-Intervention Agreement proposed by the British, and therefore cease arms sale to both sides, gold continued to flow to Paris in order to buy hard currency with which to purchase arms elsewhere. These transfers continued until March 1937, by which stage just over a quarter (26.5%) of Spanish gold reserves were now held by the Bank of France.

The controversial part was what happened to the rest of the gold. The Soviet Union had decided to support the Republic directly in September 1936, chiefly as a response to continued intervention by Fascist Italy and Nazi Germany. Most of the remaining gold - about 4/5ths of the amount shipped to Cartagena, which excluded some of the amount already transferred to France - was shipped to the Soviet Union in late October. The total value of this shipment was just over $500 million at the time, though this excluded, as noted above, any artistic/numismatic value, which is one source of the aforementioned controversy. Calculating the value of the arms provided (some estimates conclude that the Republic was routinely overcharged by about a quarter for weapons), plus shipping and other legitimate expenses incurred by the USSR, is also difficult, and continues to fuel the debate about whether the Spanish Republic was cheated.

This gold was used in several ways. Part of it was used to pay the substantial (c. $50 million) debt already incurred in purchasing Soviet arms. Over the course of 1937, about half of it was liquidated for hard currency to purchase arms elsewhere, and another $131 million was used to pay for further Soviet arms shipments. Eventually, however, the gold ran out, and the Soviet Union agreed to grant an initial $70 million line of credit to purchase more arms (which was quite a negotiating coup for the Republicans, who expected to get much less) in February 1938. This, in turn, was extended further in late 1938, though there is less clarity as to by how much. This - combined with the willingness of the Soviet Union to send arms before receiving any gold shipments in September-October 1936 - is taken as evidence that Stalin's motives were not entirely mercenary, though there is little doubt that the first arms shipments were made in the expectation that the gold would soon be forthcoming. Given that the loans would never be repaid after the defeat of the Republic in early 1939, it is difficult to conclude that the USSR profited hugely from the conflict. But, whether or not they got fair value for it, this is how Spain's bullion reserves ended up getting used.

Anyway, I appreciate that this is a bit tangential to your main question, and I hope someone with broader knowledge of bullion flows between the Americas and Europe can fill in the rest of the story.

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Sources

I've mostly followed the relevant chapter ('Spanish Gold') in Daniel Kowalsky, Stalin and the Spanish Civil War ( Columbia, 2004), which in turn draws on the research of scholars such as Angel Viñas, Gerald Howson and Burnett Bolloten.

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