Bechuanaland was pretty much joined at the hip with South Africa until its independence. Them staying in the RSA might have a big impact, as Seretse Khama made a big impact on the ideas of Britain's lawmakers, a fact which horrendously pissed off the apartheid state and wound up with a bunch of egg on Clement Atlee's face in 1948-49. Having Bechuanaland be part of the RSA means that Khama now is influencing the government in Pretoria. Assuming the strict segregation of apartheid doesn't come to pass (The United Party had plenty of chances to do it in 1910-1948, never did), then you could probably have Khama play a role in the politics for the future.
As far as SWA goes, South Africa could have annexed it easily, but it would have to be long before SWAPO comes into the picture. That said, without the staunch segregation of apartheid, that might be easier for the world to accept. The two were effectively joined at the hip by 1960, and that state exists to this day.
Rhodesia is harder, simply due to divisions. The whites of South Africa were divided between the Afrikaners and those of English-speaking descent (which includes the UK, Ireland, Australia, Canada and the USA), a divide that largely fueled South African politics. The Afrikaners would be very, VERY opposed to Southern Rhodesia being added to the RSA, which is why even if that referendum had passed in 1923, I can see the RSA trying to stall or take it easy on the integration of Southern Rhodesia, just so they didn't piss off the Afrikaners too much.
Now, assuming the United Party doesn't come to power in 1948 and you have the RSA absorbing the SWA and Bechuanaland, then the RSA has a dominant position in the world's strategic minerals market. Assuming the economic boom of the 1950-1975ish in the RSA goes ahead, then it would be added to when the diamonds of Jwaneng and the Namibian deposits are found. Few know that the unemployment of the RSA sank into the 5% ballpark by 1970. Now, that assumes that few women are in the workforce, which is the custom in both many of the black tribes and the Afrikaners. You'd likely have the same result here, if anything the wealth of the whites would be by the late 1960s start spreading to the other races.
Now, assuming the United Party stays in power in 1948 and Jan Smuts dies in 1950 as in OTL, the United Party would end up with a major division crisis, which would likely be added to by the likes of Nelson Mandela, Oliver Tambo and Seretse Khama. The Afrikaners would almost certainly demand a new leader who would take a harder line on racial relations, but this is unlikely - the Afrikaners primarily voted for the NP, so the United Party might take a easier line.
By the late 1950s, calls would start coming up for the first black members of the government. (They started showing up in the black British Empire colonies about this time.) Guys like Mandela, Khama, Tambo, Albert Luthuli, Joshua Nkomo and Robert Mugabe would be at the front of this line. This would likely draw a rift between the Afrikaners and the Uitlanders.
Black government members grow in numbers in the 1960s. The fast growing economy of the country in the 1960s, including some big discoveries (Jwaneng was discovered in 1967 in OTL) provide more employment and hard currency for the RSA. Heavy industry has now started showing up in mass amounts in the RSA. At the time, many of the laws that kept black people down specifically would start getting repealed. The ANC is at the forefront of this, and without the massacre at Sharpeville (which is much less likely in this TL) and the beginnings of Umkhonto we Sizwe, the racial peace is much clearer.
The United Party is likely to keep immigration going strong. The National Party was more concerned with diluting the Afrikaner base than with immigration, which means that plenty of immigrants land in the RSA in the 1950s and 1960s, many from the British Isles. As the black African states become independent in growing number through the 1950s and 1960s, many immigrants would come from there. The end of white rule in Congo and Kenya has a major impact on the RSA's politics, as the nation, while not finally starting to accept blacks in politics, is stridently, almost viciously, anti-communist.
By 1970, The RSA's white populations live by first world standards, and a growing number of educated black Africans are also entering that lifestyle. The last laws on real division of the races are by now dying away. Laws with regards to voting would probably spread to those with certain levels of education, which results in a major growth in the number of voting-eligible blacks who begin exerting influence by the mid to late 1970s. And without the sanctions and wars in Angola or the Bush Wars, there is more money for the RSA to have on economic development. Portugal's give up of Angola and Mozambique in 1975 results in some 350,000 new arrivals from the Portuguese colonies. The newest arrivals, owing to the extreme distaste of communism, come from parts of Eastern Europe.
By 1980, blacks are 40% of the voters, a number expected to hit 80% by 1995. Despite this, white immigration continues. The RSA's white population tops 6 million in 1980, and is anticipated to top 7.5 million by 2000. While the education and social systems are still very unequal, that is improving rapidly. The "South African Industrial Heartland" stretches out in all directions from Pretoria, including the mines and factories of the Witwatersrand, and the mines of southern Botswana, and stretching north through the towns and cities such as Pietersburg, Beitbridge, Bulawayo, Gweru and Salisbury. By now, suffrage has encompassed virtually the entire Indian and Colored populations. In fact, the Indian population is growing as people come to the RSA from India.
The United Party finally loses power to the ANC in 1986, after the longest uninterrupted reign in a modern democratic state with real and fair elections. The RSA is ardently anti-communist, to the level of being a major ally in the Southern Hemisphere for the western powers. The ANC, which has always fought for universal suffrage, advocates a rework of the constitution to allow for universal suffrage, gets its wish in 1990. The 1992 South African elections are the first universal suffrage elections. The ANC wins, but the UP provides a strong showing - helped by a large number of black candidates and the South African blacks being a lot less uncomfortable with the idea of white members of government.
Despite universal suffrage, the white lion isn't done yet. The collapse of the USSR sees a new wave of white immigration hit the RSA. True to form, the white population of South Africa hits 7.5 million in 1995.
2009 South Africa is by far the most prosperous place on the continent, a fact which provides both benefits and problems. The RSA faces major problems with illegal immigration, most notably from civil war torn Mozambique and Angola. The RSA has a $1.24 Trillion economy for a country of 64 million people, which puts it on an equal footing with many nations of the West. This includes 7.8 million whites, 4.2 million coloreds and 1.6 million Indians. Black Africans are the overwhelming majority, of course, but the blacks of the RSA live better than any other African nations by far. The smartest ones at first world standards.