I study U.S. foreign relations in southern Africa, so I can speak to a fair bit of this.

On paper, no country officially recognizes the Rhodesian UDI for the simple reason that it sets an incredibly dangerous precedent. Not a unilateral declaration of independence per se, as those have existed throughout history, but the fact that this country could declare independence from another power while denying majority rule to its citizens (as no government really buys the notion that whites did not predominate in Rhodesian life in every meaningful way). Britain had established that none of its colonies could acquire independence without granting majority rule to black Africans after WWII, and so there was a legal precedent by which the UDI could be ignored.

However, and this is the critical point, there were states or groups within states that covertly supported Rhodesia. I'll break them down here.

Portugal and South Africa. Portugal treated its African colonies as integral parts of Portugal itself, and Angola and Mozambique, its two largest African possessions were experiencing guerrilla insurrections by 1965. South Africa's policy of apartheid meant that it was also trying to enforce similar racial mores and was furthermore looking for allies, as well as a way of maintaining its control over Namibia, which also fit into this southern cone. As many of the liberation movements of those countries had fled north into the sympathetic states of Tanzania and Zambia and thus cooperated, this meant that in turn the white minority regimes cooperate militarily against their respective insurgencies. This is formalized in 1970 with what is called the Alcora (acronym translated from Portuguese: Alliance against the Rebellions in Africa) Exercise, under which they commit to providing assistance to one another. Portugal keeps diplomatic personnel in Salisbury (Harare) until 1970, and the Rhodesian mission in Lisbon remains open until 1975.

This military assistance is significant because it means that Rhodesia is effectively ensconced in a cordon of allies, who help to maintain control of the country's borders and stop aid from reaching the Zimbabwean insurgent groups, ZAPU and ZANU. This is partly why the Rhodesian Bush War doesn't really heat up until after the Carnation Revolution in Portugal removes the dictatorship, because Portugal takes some of the heat off of Rhodesia.

They also, and this is critical, keep trade flowing out of and into Rhodesia. Zimbabwe is a landlocked country that depends on its neighbors for access to the sea, and in the wake of the UDI, the UN enacts a trade embargo that theoretically should have crippled the government. However, active support from the Portuguese and South Africans means that vital goods such as oil flow into Rhodesia, while Rhodesian exports such as tobacco and chrome flow out of the country through those same powers. They do this with a fair amount of support from western corporations; Mobil Oil for example worked with a series of South African shell companies to permit petroleum to reach Rhodesia.

I wouldn't say that Rhodesia attracted majority support from Americans or any other state that actually practiced majority rule, but there were definitely groups that aided, supported, or just generally rooted for the Rhodesians. In the United States, you see groups such as the Friends of Rhodesia spring up, and while they were carefully cultivated by the Rhodesian Information Office, they did find Americans who genuinely believed that Rhodesia was a viable state worthy of support. Nor was this confined to a racist fringe in the American South, which is the stereotype. One of the most prominent supporters of Rhodesia was Dean Acheson, the former Democratic secretary of state, referred to Rhodesia as a beacon of western civilization; the last letter written before he died in 1971 dealt in part with the abandonment of Rhodesia as he saw it.

In Britain, where ties to Rhodesia were much stronger, there was open support for white minority rule in the Conservative Party. Members of the Monday Club, a conservative group (and made up of members of the Conservative Party) backed Smith. Nominally, many people used the excuse of anticommunism because the Rhodesian government claimed to be resisting communism, but in practice, there was a combination of white supremacy, "western civilization" chauvinism, anticommunism, and economic opportunism that leads to support for Rhodesia in the west.

This culminates in the United States in the Byrd Amendment, which was sponsored by Harry Byrd of Virginia prohibited the U.S. government from banning the importation of materials from a non-communist country if in turn it wasn't banned from a communist country. While not explicitly about Rhodesia, in practice, it was a way to allow Rhodesia to export its chrome to the U.S. (the other major stockpile came from the Soviet Union). This was framed as a Cold War issue because it lessened U.S. reliance on imports from the USSR, but it also legitimized the Rhodesian government in the eyes of some, weakened the reputation of the U.S. in most of Africa, and gave Rhodesia some foreign currency as well as a morale boost.

Lastly, there's the issue of white mercenary support for Rhodesia. At its peak, Rhodesia never numbers above more than 300,000 whites in the country (the number fluctuates continually), and even with draft requirements that heavily target both whites and the Indian minority, they cannot raise enough troops as time goes on. This leads them to hire foreign mercenaries from abroad, many of whom come from the United States, Britain, Canada, and other Anglo countries. U.S. veterans, many of whom served in the Vietnam War, are targeted through publications such as Soldier of Fortune. The historian Gerald Horne alleges that many of the mercenaries who were hired in the U.S. themselves had racist sympathies and an implicit tendency to support the Rhodesian government.

Further reading... Gerald Horne, From the Barrel of a Gun: The United States and the War Against Zimbabwe (A problematic book because of source use, but a worthwhile perspective nonetheless)

Nancy Mitchell, Jimmy Carter in Africa: Race and the Cold War

Susan Onslow, Britain and Rhodesia