For those of us who haven't seen the movie, what events are they referencing as potential catalysts for a broader war?
this would be a stretch from my perspective, however you'll have to let me get ahold of my notes from my Master's research so I can give you a bit more to chew on. my focus was on the Johnson period, specifically on the transitionary policies leading to Mobutu's rise.
I'll offer these opening points though:
the only hint of a potential global scale conflict emanating from Soviet/US policy and/or UN policy is in Lise Namikas' Battleground Africa, where she outlines a conflict that was entirely chaotic for policy makers in Washington. for her, the overreaction in Washington to perceived Soviet aggression escalated the conflict from mere political crisis to full-blown civil war. for her, from 1960 forward, the US eschewed a multilateral approach in favour of direct confrontation with the Soviet Union. this could be seen as a possible starting point for a WWIII along the lines of Soviet-US divisions.
there's no indication that the Soviets were really willing to dive deep into the Congo Crisis after 1962. they tried lobbying governments (particularly Cyrille Adoula's) in Leopoldville but were fairly silent otherwise. this might have changed should the US have really lost control of political elites in the Congo, or if Tshombe had shown a real willingness to confront the UN and NATO/US military power. for that matter, should the US have shown a real willingness before 1962 to intervene militarily in Katanga. either way, I would highly doubt that the Congo Crisis would have erupted into full blown WWIII, especially with American/Belgian mining companies lobbying for business as usual (Stephen Weissman is useful here).
the reality is, the Cold War was an era where rumours of global conflict were constantly bandied about, though I would argue mostly in popular discourse rather than political discourse (excuse my broad and generalized analysis here). this could very well be one example of that creeping into the movie's dialogue.
I'll try to edit this for more comprehensiveness in the morning, though it is the first day after the holiday break so it will be a busy one!
edit:
having reviewed some of my notes and having delved back into Madeleine Kalb's Congo Cables a bit, I'm just going to reiterate that any hint of a possible WWIII would constitute a great exaggeration. sure, the CIA and the KGB were heavily involved in trying to lobby favourable outcomes but after Lumumba's assassination in early-1961 the conflict was basically diplomatic outside of UN involvement until the Johnson Admin. neither the Russians nor the Americans really wanted to get too involved in the Congo Crisis except at an arms-length. from time to time, especially in dealing with Tshombe/Katangan secessionists, the possibility for any direct confrontation between the super powers was there but quite limited. though funding for Tshombe and Antoine Gizenga (a Lumumbist supported by the Soviets) was a constant worry for American and Soviet policy-makers, multilateralism remained the focus of policy until the Johnson administration (who bought into the Kasa-Vubu/Mobutu option pretty heavily by late-1965). further, as most of the scholars I've mentioned have noted, Khruschev - though initially quite involved in Congo policy - did not see Africa as a viable option for a challenge to US power after Lumumba's assassination. so to directly answer your question: the claims don't seem to be accurate at all, at least from an American foreign policy perspective.
The way that the Cold War enveloped Congo in 1960 and 1961 was not militarily, but politically. Both superpowers attempted and succeeded in shaping the political situation – for example the Soviet Union supported prime minister Patrice Lumumba both financially and militarily (although the extent of the military aid should not be exaggerated, mostly it was transport planes, trucks and weapons). The US, on the other hand, financed and supported Lumumba’s opposition, including a young colonel by the name of Joseph Desire Mobutu. US policy in 1960 and 1961 was about containing the crisis and to prevent Soviet-supported politicians like Lumumba any influence. Some CIA sources even show that they planned to assassinate Lumumba, although the Belgium-supported Katangese did so instead in February 1961. US wanted a moderate, pro-western government and that was established in July 1961.
Archival studies seem to indicate that neither the US nor the Soviet Union wanted an escalation of the crisis. What we do know is that there were some US policymakers and within the CIA who saw Congo as a potential Cuba (and Lumumba was often likened with Castro). (Kent p. 28) However, Allen Dulles who was the director of the CIA, acknowledged in 1962 that they had over exaggerated the Soviet Union’s influence. (Meredith p. 113) The deputy ambassador to the UN, Charles Yost, argued at the time that “it was doubtful if the aim of the Soviets was to establish a communist regime in all, or part, of the Congo.” (Kent 48) How Congo compares to other events during the Global Cold War and the doctrine of domino theory, I cannot say.
A few weeks after independence, the Pentagon put two companies of the U.S. Twenty-Fourth Divison in West Germany on alert to “help evacuate refugees from the Congo if it became necessary”. (Kalb p. 9) That is about as close as the US came to deploying military units in Congo until Operation Dragon Rouge in 1964. In 1964 and 1965, after the UN left Congo, there were proxy-fighting in Congo during the Simba rebellion. I haven’t studied much of this, but if you’re interested I’d suggest reading up on the rebellion, as well as Operation Dragon Rouge. Che Guevara makes a surprise appearance before Mobuto crushes the opposition and holds on to power until 1997.
In conclusion, the clock was not five minutes to midnight, unlike what a movie may say for dramatic effects, especially in September 1961. Neither USA nor the Soviet Union were interested in war, rather they used their diplomatic and political influence to try and shape Congo in a way that suited their geopolitical interests.
Regarding uranium in Jadotville (in Katanga), I have not seen any sources or historians claim that it influenced Soviet or US policy at all. In fact, I doubt that Congo was the only place where the superpowers supply of uranium came from. It’s probably just a way for the movie to add some dramatic tension, it would be disingenuous to suggest that Katanga’s uranium was in any way connected to the events.
It may be worth noting that the Second Congo War has been described as "Africa's World War". In a similar way to how WWI started, mutual defense agreements and changing alliances led a bunch of different countries to send troops to fight in DRC. So if you are thinking of "World War" as a regional conflict that expands to pull in countries from all over a continent, then it kind of, sort of did happen, just much later on.
I don't want anyone to be confused because the Katanga Crisis happened 35 years before the Second Congo War, so the political and international situation was very different. However, I can't imagine the film makers did not know about the Second Congo War and it would have given them a clear understanding of how a conflict in Congo could have escalated and spread to other countries throughout the continent. I don't want to go into speculative history here, but the threat of an escalating conflict in Congo causing a greater (and even nuclear) war between the USA and USSR is at least plausible enough to include in the movie as something the characters would be worried about.