Regarding Italy, the scenario looked plausible in 1977, but not necessarily likely. u/Alvisefalier has some background on the PCI (Italian Communist Party) here.

1977 in particular came on the heels of the 1976 Italian elections, which saw the PCI come in at a close second behind the Christian Democrats (34.37% of the vote and 228 deputies out of 630, compared to the Christian Democrats' 38.71% and 262 deputies). There was a widespread fear among non-communists that the PCI would form a government and change Italian foreign policy stances, but this never really was a likely outcome - other Italian political parties formed a government with the Christian Democrats, and the PCI ended up providing external support briefly, before returning to the opposition (and lost seats in the 1979 election after the crises involving terrorist attacks by the Red Brigades in 1978 and 1979).

For what its worth, from the late 1970s Western European communist parties (including PCI) adopted "eurocommunism", which was stance that largely distanced themselves from the Soviet Union and Soviet foreign policies. Interestingly, the French Communist Party (PCF) consistently got over 20% of the vote in legislative elections in the 1960s and 1970s and ended up contributing ministers to the French Government in 1981-1983 (despite losing seats and votes heavily to the Socialists in 1981), and this didn't really change French foreign policy much under Mitterand.

Depending on the details, the African example....sounds kind of like what actually happened? The Marxist-Leninist MPLA formed a pro-Soviet government in Angola, and was supported with 36,000 Cuban troops in its civil war against UNITA (which was backed by South Africa, which invaded Angola in 1975). Mozambique likewise achieved independence under the Marxist-Leninist FRELIMO (and received substantial support from the USSR and Eastern Bloc), which became embroiled in a civil war against the South African-supported RENAMO. Zambia, Botswana and Mozambique also provided refuge and material support to the African National Congress (ANC) and the South West African People's Organization (SWAPO), and this led to South African commando raids in these countries in the 1980s.

As for India disintegrating, the 1970's were the period when it wasn't clear whether more post-Colonial states would break up into smaller states based on clearer ethnolinguistic divisions.

Decolonization was mostly over. Even the Portuguese colonies in Africa — Angola, Mozambique, and Guinea-Bissau — had decolonized in 1974-1975 after the Carnation Revolution in Portugal and a long bloody war in the colonies. (To address another issue, many of those anti-colonial struggles, especially the anti-colonial opposition had been supported by the Soviets—check out Angola's modern flag, instead of the hammer and sickle representing the industrial and agriculture workers respectively, they chose the local equivalents, the gear and the machete.) There had been a big swing, more or less starting with India, of (on a world historical scale) rapid decolonization through Britain, France, Portugal, and other colonial power's now former colonies.

There were lots of questions for these new states (Capitalist First World, Socialist Second World, or non-aligned?; development?; national unity?; minority policy?), but one of them was what to do about these inherited colonial borders. As you may know, a lot of these borders just make no sense from an ethnolinguistic/ethnoreligious perspective. There was a lot of debate about what to do in this post-colonial period. Syria and Egypt joined together as the United Arab Republic from 1958-1961, which Yemen also sort of joined briefly, before the whole thing came apart. Tanganyika and Zanzibar were colonially administered separately, gaining independence in 1961 and 1963, but joining together in 1964 and remaining together even today as the modern state of Tanzania. Malaysia formed as a union of the British Federation of Malaya colony in mainland Southeast Asia united with the separately administered insular colonies of Singapore and British North Borneo and Sarawak in 1961 (they had all been granted independence or self-governance between 1957-1961) only for the rest of mainly Muslim (with a large Chinese minority) Malaysia to kick out mainly ethnically Chinese (with a large Muslim minority) Singapore. So these borders really were getting figured out.

One of the things that I assume that was on General Hacklett's mind at this period was Biafra. Nigeria is one of those odd post-Colonial states, roughly half Muslim and half Christian/animist, with each ethnoreligious identity breaking down into smaller ethnolinguistic groups. The largest four groups are the mostly Muslim Hausa and Fulani in the North and the mostly Christian/animist Igbo and Yoruba in the Southeast and Southwest respectively The Biafran War (1967-1970) was an attempt basically an attempt to form nation-state for the Igbo ethnic group separate from the rest of Nigeria. Nigeria had only actually became independent in 1963. I like this sentence in Wikipedia, "Immediate causes of the war in 1966 included a military coup, a counter-coup, and anti-Igbo pogroms in [Muslim-majority] Northern Nigeria" because it gives a sense of how power was being figured out in this diverse country. If the Igbo won independence as Biafra (Biafra would be about 60-70% Igbo, 30-40% non-Igbo mostly Christian/animist groups like Ibibio, Ijaw, Ikwerre, Ogoni), then probably would have necessitated the Yoruba and other mostly Christian/animist groups to break away as well. The Biafran War was bloody, with roughly 100,000 killed in combat, but more memorably it created a huge and devastating famine that killed between 500,000-1,000,000 people and may well have created the "there are starving kids in Africa meme" (before that it was China, before that it was Armenia — see my old answer here). Images and stories of starving children were everywhere in Western media and support for Biafra became a cause célèbre in the West.

The Biafra War for Independence failed and Nigeria has managed to endure as a multi-ethnic, multi-religious state (with periodic communal violence). Even though this particular attempt failed, it was not unreasonable to wonder if this was going to keep happening, After all, World War I and World War II were essentially started about would the nation state exist in Europe. Would these post-colonial states break up into more clearly national states, with France for the French, Poland for the Poles, and Igboland for the Igbos, Yorubaland for the Yorubas?

The one time we'd already seen this splitting of a major colony along communal lines was in British India, which split in India and Pakistan in the Partition in 1947, with millions of Hindus and Sikhs going one way and millions of Muslims going the other. 10,000,000–20,000,000 people were displaced and 200,000–2,000,000 were killed in communal violence around Partition. Also probably on General Hackett's mind was the recent (1971) Bangladesh Liberation War, where mostly Bengali-speaking East Pakistan broke off from officially-Urdu speaking West Pakistan. The War was short, but violent, with roughly as many people dying as had died in Partition. Similarly in the early 1970's you saw Black September (a war between Jordanians and Palestinians in Jordan), the beginning of the Lebanese Civil War where the country divide mostly on ethnoreligious lines, and, though not in a Post-Colonial context, the Troubles in Northern Ireland again on ethnoreligious lines. Could all post-colonial and multiethnic states dissolve through bloody conflict into smaller ethnolinguistic units?

[continued below]

Just to complement the post by u/Kochevnik81 with some more examples and ruminations:

--In the 1970s the implantation of communism in Jamaica was quite plausible.

--Eurocommunism actually plays a role in the novel besides explaining Italy's failure to fight.

In 1972, the left-wing People's Nationalist Party had come to power for the first time since independence. Michael Manley of the PNP was prime minister. Manley pursued close contacts with Cuba, supported anti-imperialism and anti-apartheid in southern Africa, and spoke favorably of the Soviet Union. So communist Jamaica was somewhat plausible at the time the book was written--and was enough of a concern that the CIA was destabilizing Jamaica, which saw a surge in political violence during the 1970s.

What actually happened to cut this short was in 1980 the right-wing party (confusingly called the Jamaica Labour Party) won in a landslide. During the 1980s Jamaica was led by the JLP's Edward Seaga and aligned with the United States and United Kingdom. So for example real Jamaica supported the invasion of Grenada in 1983 and was the main anti-communist country in the Caribbean. But it could have come out differently.

On the other hand, Grenada had become communist in 1979 thanks to a coup by the New Jewel Movement which installed a revolutionary government under Maurice Bishop. Bishop's execution in 1983 by harder-line Marxist-Leninists (Bernard Coard, Hudson Austin, and the People's Revolutionary Army) was the precipitating factor for the American-led invasion.

So communist Jamaica wasn't all that far-fetched in the mid-70s, and if the prediction were merely, "A Caribbean country will go communist in the next decade," you'd probably guess it would be Jamaica.

The other key political development, though, is Soviet realization their hold on Eastern Europe is weakening and Eastern Europeans may see a path forward as Europeans rather than Soviet subjects. Here the novel's ambivalence about eurocommunism is clear: while eurocommunist Italy crumbles under the stress of war, on the other hand the very existence of eurocommunism as a socialist alternative to Soviet domination is part of what makes the Soviets risk war--and the Soviets of course lose that war. Indeed, rereading a couple times since 2000, I was struck by how much Third World War is about European integration.

On the large question of plausibility v. "moving the pieces," it's really a mix. Hackett was strongly influenced by an earlier book calling for NATO rearmament, L'Europe sans defense ? by Belgian general Robert Close. Close argued that NATO was very vulnerable to a surprise conventional attack by the Soviet Union and spun out a developed hypothetical (not exactly a fiction treatment) showing how such an attack would unfold. Basically, the Soviets launch a conventional attack straight from their barracks on a weekend and move so fast NATO (many of whose troops are on leave and whose vehicles, aircraft, etc. are not fueled or armed) can't prevent deep penetrations, so NATO has to choose between civilization-destroying nuclear war and accepting Soviet demands. Close also attempted to respond to some possible objections, one of which was that the Soviets had no reason to do something so risky. His response was essentially that the political situation could change and the Soviets could decide to risk war.

Hackett's goal in The Third World War was to sound the alarm about NATO's conventional weakness even more forcefully. So part of what Hackett and his team did was create a situation in which the Soviets choose to risk war. And he does so in a way that was pretty plausible at the time but whose details--except for losing grip on Eastern Europe--are not all that important to the story. This is pretty much borne out by early reviews, which talk about the main story of the war and the plea for rearmament. After the fall of the Shah, reviewers start commenting the political story isn't quite right.

The authors eventually concluded they needed to update the story and published a sequel in 1982: The Third World War: The Untold Story. This volume retcons (as we'd say now) the story so it more-or-less fits with actual history (fall of the Shah, Reagan elected, Solidarity). It didn't sell as well, perhaps because it's even less of a novel than the original and also doesn't really tell the main story of the war. (Within the genre, I'd say the 1982 volume did have a big influence since it contained sections focusing on the experiences of Soviet authors, something that became more and more common in the conventional WWIII genre in the 1980s culminating in two 1989 books written from the Soviet perspective: Ralph Peters, Red Army and Steven Zaloga, Red Thrust.)