The eradication of smallpox required global coordination among hostile geopolitical rivals at the height of the Cold War. How was this accomplished?

Quick disclaimer here - I'm an epidemiologist, not a historian.

My sources here are: Randall Packard's 'A History of Global Health', DA Henderson's, 'Smallpox: the Death of a Disease', and William Foege's 'House on Fire: the Fight to Eradicate Smallpox'.

Before getting to your specific question about the politics of eradication, I want to do the obligatory epidemiologist thing and point out that there are a number of biological reasons why smallpox eradication was successful: the disease had a number of characteristics that made it a good target for such an attempt. It infected only humans (no animal reservoir to lurk in), a single infection confers high immunity, the symptoms are highly characteristic (making it easy to find and trace cases) and it is relatively less infectious than diseases like measles and chickenpox (making it easier to interrupt transmission). You can kind of say that a number of key things worked together to make eradication possible: one were these characteristics of the disease, another was the political will. Foege also emphasizes a few big players who pushed at the right moments (and put politics aside at a few critical moments), and a lot of on-the-ground workers who committed themselves to the goal of disease eradication.

So, according to DA Henderson: in 1953, Brock Chisholm, the Canadian director of the WHO, proposed a smallpox eradication campaign and was summarily rejected by the world health assembly due to concerns about cost, among other factors. The campaign only got off the ground when it was proposed again, by, guess who? - the Soviets! - in 1958. This was a big deal because it marked the USSR's first engagement with the WHO since their withdrawal in 1949. The USSR was interested in smallpox because they had largely managed to halt transmission within their own country by the 1930s (even with limited health-care infrastructure and not a very good vaccine, which again shows that the smallpox was a good target for eradication) but were plagued by regular re-introductions from neighboring countries.

From 1959-1966, a bit of a cold-war standoff emerged between 2 competing major international health campaigns: smallpox and malaria. Smallpox eradication was the USSR's idea, and the WHO supported eradication, but only modestly (in financial terms). The US, in contrast, was heavily financially invested in malaria eradication, which was not going well, and also argued that WHO money should be limited to providing technical support to smallpox, and should not be used to purchase the vaccine.

In 1963 and 1964, however, the US's position on smallpox was changing. Henderson describes this as the result of two big smallpox scares, which led the CDC to start thinking seriously about how to vaccinate large numbers of people in a hurry, should an outbreak occur. This led to the introduction of a jet-injector gun system that could vaccinate a thousand people per hour. Then, in 1965, the CDC came up with a plan to piggyback smallpox eradication onto a USAID measles campaign in West Africa as a way propping up the measles program- President Lyndon Johnson was unexpectedly enthusiastic about this idea. The funding for the program then had the effect of revitalizing the flagging WHO smallpox efforts.

During the same period, the WHO's membership was rapidly changing. According to Packard, in 1951 the WHO had 51 voting members, mostly in the US and Europe. By 1965, there were 121 represented, many of whom were newly independent countries that were majorly affected by smallpox and were motivated to deal with the problem. When a new smallpox proposal was brought to the WHO (by Henderson, who had just come over from the CDC), the US and other donor countries wanted to slash the budget, but a block of developing countries from Africa, Asia, and Latin America pushed it through.

TL;DR- the USSR was on board because it was their idea, the US got on board when they realized that malaria eradication wasn't going to work out and they were looking for ways to prop up their international health program, and many other countries were also on board because smallpox was a major problem.