Born on the battlefied and sired by an illegal white-minority government, the new black-led government of Zimbabwe-Rhodesia will be in a highly precarious position when it takes office later this month.

The government's chances of gaining the international recognition it needs to survive are uncertain, even though Prime Minister-elect Bishop Abel Muzorewa maintains that the 64 percent turnout in last month's first universal suffrage elections gives him more legitimacy than most african leaders.

No black African state has even hinted at recognition. The Organization of African Unity has condemned the elections and called on its members to stand by the Patriotic Front guerrillas fighting the government.

The new Conservative government in Britain is already being pressured by members of the Commonwealth not to extend recognition amid warnings of its dire consequences. In reaction, the Tories, while clearly leaning toward recognition, have already indicated that no formal decision is likely until after the Commonwealth conference in Lusaka, Zambia, in early August.

In the coming months, a series of conferences is scheduled to take place in Africa that likely will shape the diplomatic fate of the new government in Salisbury. French-speaking African states met last weekend in Rwanda with French President Valery Giscard d'Estaing. In July an OAU summit will he held in Liberia, followed by the meeting of former British-ruled countries in Zambia.

Britain and the United States in particular are likely to monitor these gatherings closely to see whether the diplomatic winds in Africa are supporting the Muzorewa government.

By the end of the summer, it should be clear whether Muzorewa will win his struggle for African recognition. This in turn is likely to influence London's and Washington's decisions of whether to recognize Zimbabwe-Rhodesia, even if the U.S. Congress forces the Carter administration into lifting economic sanctions as it now seems to be in the process of doing.

Zimbabwe-Rhodesia may then emerge as a fait accompli on the explosive southern African scene, albeit something of a pawn in the hands of the regional white economic and military colossus, South Africa. Or it could prove to be nothing but a sand castle destined to be swept away in the rising tide of strident black nationalism throughout the region.

Rhodesia is presently swamped with about 12,000 guerrillas, whose leaders and outside backers are determined to destroy the Muzorewa government. The war continues to escalate steadily. Already the official death toll reached more than 14,000 and probably close to 20,000.

Ninety percent of the county is under martial law and no main highway or city is safe from guerrilla attack. Thinly stretched Rhodesian security forces have all but given up control of a significant if undeterminable portion of the countryside, particularly along the eastern and western borders.

Privately, white Rhodesian authorities admit that the war cannot be won on the battlefield alone and that the success of the white-backed Muzorewa government hinges on securing outside support. This explains the increasingly desperate pleas issued almost daily by the bishop and Smith and echoed by their lieutenants for the international community to grant diplomatic recognition and lift economic sanctions.

If neither is forthcoming, Muzorewa and everything he stands for are certain to end as little more than interesting relics of Rhodesia's white and black nationalist histories.

But if the new British Conservative government decides to recognize the Muzorewa government, a handful of African states follows suit and the United States lifts economic sanctions, the government has at least a fair chance of defeating the guerrillas and surviving.

The stakes are enormous, not just for Muzorewa and Rhodesia's dwindling white population, but also for South Africa, the West and the East. This highly, prized piece of real estate, rich in minerals and an agricultural powerhouse, has the potential of becoming the next battleground in the East-West struggle for influence across Africa.

Rhodesian policymakers would love to turn Zimbabwe-Rhodesia into just such an East-West contest and are using every means to get the United States to take a stand against the encroachments of the Communist world here.

Thus, the Rhodesians constantly harp about direct Soviet involvement in the conduct of the guerrilla war from Zambia, although authorities will provide no hard evidence of this and even the Chinese in Lusaka doubt it.

The top priority for Rhodesian strategists is to break the Carter administration's opposition to lifting the economic sanctions or else to navigate an end run around it.

"We're going for sanctions lifting. Once that's achieved, the rest will follow," remarked one Rhodesian who was highly optimistic about achieving this goal.

Rhodesians have been encouraged tremendously by growing bipartisan support in Congress for the Zimbabwe-Rhodesian cause. They are busy laying plans for a squeeze on the reluctant Carter administration, counting on the new British government on the one hand and their conservative supporters in the United States on the other.

These strategists calculate that if Congress votes to lift sanctions, the action will have a catalytic effect on the Tory government in Britain and resolve the agonizing dilemma of which Western or African country moves first to break the diplomatic logjam holding up recognition of the Muzorewa government.

South Africa, in turn, cannot but reap enormous benefits from a break in this logjam. It has become clear recently that Pretoria intends to use the white-backed Muzorewa government as its northern bulwark against Soviet and Cuban-supported black nationalists battling inside Rhodesia and already beginning to stir within South Africa.

For its part, Pretoria has been promoting a plan of regional cooperation that would help control guerrilla incursions against its white-minority government. South Africa's goal is a web of moderate, black-governed buffer states friendly to it, and Muzorewa's government plays a key role within that scheme.

Observers in South Africa say that unless the West shows a willingness to trade with and recognize the new Salisbury leadership, Rhodesia will be forced into closer cooperation with Pretoria's regional plan.

The South Africans appear prepared to use economic leverage and behind-the-scenes diplomacy to support the Muzorewa government. Observers feel, however, that Pretoria probably will not formally recognize the Salisbury leadership until some of the moderate black states do.

"South Africa and the bishop will have a quiet relationship for some time," one observer said. "They don't want to kiss him to death."

South Africa, however, provide vital military assistance to Rhodesia, supplying ammunition and providing a means for Salisbury to circumvent U.N. economic sactions so that it can import other equipment.

One source said South African Army officers are allowed a year's leave of absence if they want to serve in the Rhodesian forces, after which they can return with their seniority intact. South African liaison officers are sent into the Rhodesian bush for months at a time and they are unconfirmed reports that South African police operate regularly against guerrillas in southeastern Rhodesia.

According to other unconfirmed reports, South Africa has supplies Rhodesia with a squadron of Mirage fighters.

With South African and Western backing, strategists of Zimbabwe-Rhodesia's survival are confident that they can win over or crush the majority of guerrillas.They feel a dozen or so moderate black African and Arab states will follow if the West takes the lead in recognizing the Muzorewa government and that the effect of this would be to isolate the front-line states of Mozambique, Botswana, Zambia, Angola and Tanzania backing the Patriotic Front guerrillas. Eventually, they argue, assistance for the guerrillas even from their closest allies would wither away.

There is a huge gamble in all this for the United States. A commitment to Zimbabwe-Rhodesia might serve to halt the spread of Soviet-Cuban involvement in southern Africa or it migth give new impetus to the Communists.

What seems certain, however, is that overt Soviet Bloc commitment would posture Washington on the white side of the divide in southern Africa. For whatever the image of the Muzorewa government may be in the West, it is still viewed in much of black Africa as a wily slight of Rhodesian white hands to hold onto power beneath black gloves.

This interpretation is particularly prevalent among those black African states upon which the Carter administration has built its policy in dealing with this continent - Nigeria, Tanzania, Zambia and even to some extent Kenya.

In effect, American policy toward black Africa would revert back to what it was under the guidance of former secretary of state Henry Kissinger, tilting publicly toward the white south and away from the black north. CAPTION: Picture, Prime Minister-elect Bishop Abel Muzorewa reviews biracial honor guard at a Salisbury fair yesterday. UPI