Although the Queen's position as Queen of South Africa was distinct from her role as Queen of the United Kingdom, in the minds of most English-speaking South Africans the two roles would have been intertwined along with their own identities and British and South Africans, not unlike the other English-Canadians, Australians and New Zealanders at least until the mid-1960s. However, the Labour Government of Harold Wilson particularly alienated English-speaking South Africans, and this in turn would have caused many to turn their backs on the monarchy. In 1964, Wilson upon coming to power ended arms sales to South Africa, and his imposition of sanctions against Rhodesia greatly alienated English-Speaking whites in South Africa. If a republic did not come come about in 1961, it is likely that the South African monarchy would have had its days numbered.
Keep in mind that the Queen never visited South Africa between 1952 and 1961. During that same period she had already made visits two the other three white dominions of Australia, Canada and New Zealand, which included lengthy tours of each. Usually royal visits to the Dominions are made at the behest of the respective governments and coordinated with Buckingham Palace. King George VI had been invited to South Africa by the Malan government in March 1952, however this visit did not take place due to the king's health.
The question of course would be what would the Queen have done if she had been invited to take a tour of South Africa during the 1960s or 1970s. One could envision a scenario after the mid-1960s where the National government invites the Queen to South Africa as a propaganda coup to show that apartheid South Africa is not internationally isolated. Though Buckingham Palace could have deferred the tour due to scheduling, at some point it would have proven to be an embarrassment for the queen. For instance in 1972 she visited nearby Mauritius and could have extended her tour to include South Africa. On such a tour, she would have likely have been given a government-prepared speech to read before the South African parliament advocating "Separate Development". Keep in mind that for her to say no to her government would have sparked a constitutional crisis. In addition, any tour itinerary would have prepared by the South African government.
The question of a royal tour of South Africa by any member of the royal family would have become politicised. If the palace were to say no, it would risk offending the Queen's South African subjects and possibly sparking a constitutional crisis in South Africa by refusing to read government-prepared statements. If it says yes, then there is the issue of the Queen offending subjects in her non-white realms (Gambia, Sierra Leone), along with other African republics (ie Nigeria, Tanzania) in the Commonwealth. In effect, the Queen remaining head of state of South Africa could divide the Commonwealth.
Keep in mind that during settlement talks regarding Rhodesia in the late 1960s and early 1970s the British government did advise the Rhodesian government that it would be impossible for the Queen to become Queen of Rhodesia as it would put her in an embarrassing position, and that the country would not be accepted into the Commonwealth.
In 1961, the white governments of the "white dominions" were in favour of keeping South Africa in the Commonwealth. This was due to South Africa's wartime contribution. For instance a February 1961 Gallup Poll showed that 75% of New Zealanders favoured keeping South Africa in the Commonwealth. However, with the admission of Nigeria, the new Commonwealth members now outnumbered the old, and the African and Asian members were hostile to South Africa. Keep in mind that by the end of the decade 16 additional African/Asian/Pacific nations would join the Commonwealth, making the organisation even more uncomfortable for South Africa. This would culminate with the 1971 Singapore Declaration at the Commonwealth Heads of Government meeting in Singapore which denounced racist governments (Rhodesia, South Africa).