- The British settler colonies in Canada, Australia and New Zealand have slower movements for autonomy and independence. I have no idea what happens with South Africa.
South Africa slowly begins moving into the orbit of Germany assuming less immigration into South Africa (which I'm unsure about, on the one hand a postwar environment is conducive to people wanting to move overseas if the immigration waves of OTL are anything to come by, on the other hand, South Africa might decide to copy Australia's policy in some manner).
If stuff like the Maritz Rebellion are any example, there's a lot of pro-German sentiment among the Afrikaners during this time, and there were a lot of people itching to ditch Britain. Another thing is that the Cape will have less influence, because the rise of the Cape and of the Afrikaner business class which started up there only came around after the end of the First World War. So basically, more anglophobic Afrikaners, a somewhat less organized and smaller Afrikaner middle class, and probably a less urbanized black population (urbanization really only got into full swing after the Second World War, which I assume wouldn't happen without a first.)
All of this means that Jan Smuts's United Party stays influential for a lot longer, and Hertzog's Nationalist Party won't take power like OTL (they were bankrolled and supported by these Afrikaner businessmen, and the Afrikaner Broederbond which served as the "inner circle" of the party was started by these types.) At the same time, Afrikaner resentment might have a larger blowback, leading to something like 1948 at a later point, and with a harder blowback. I.E. worse apartheid.
This might have the consequence of alienating Anglophone South Africans, though since the balance is tipped 2-1 for Afrikaners, there's not as much that they can do except in Natal, where the vast majority of whites are Anglos. OTL after the 1960 referendum there was some murmuring in corners that things had gone too far and Natal had to secede. Wouldn't be too hard to see things OTL leading to an independent Natal, a Rhodesia-on-the-sea. And like Rhodesia, they'd likely reach universal suffrage for all its people earlier than South Africa. The IFP would probably be pretty happy.
Then again, you could go on an alternate path, something like the Rand Rebellion taking off in the Transvaal, leading to a strange Nazbol-like state popping up. (I'm surprised that no one has done a serious timeline about a
NazBol apartheid South Africa, because the idea of that is just too weird not for someone to have noticed by now. One of their slogans was literally "Workers of the World fight for a White South Africa!"
Either way, my gut feeling is that blacks will be worse off, and for a longer period of time, than OTL. I don't think you can have something like the ANC-- (which at this time was nothing more than a bunch of letter-writing, English-educated, and self-important minor chiefs' sons and activists) --without someone from Mandela's faction taking over and leading an effective insurgency campaign. There were other major groups fighting against discrimination prior to when the ANC became the most dominant force during the 70s or so, but they would have similar origins, and if not, wouldn't exist TTL.
With the white working class being poorer and in a worse state, they'd be in direct competition with blacks for jobs, and would lobby harder to slow down the use of native black workers for labor. The later ANC was deeply rooted in support from urban workers and trade unions, as well as the Soviet Union-backed Communist Party (which was headed by blacks from before the Great Depression). So without the influx of blacks into the cities, the ANC would probably be more moderate, and Mandela would be out of the picture.
This fact leads to two different things happening when it comes to apartheid policy. One would be that the homeland policy ends up being more successful, for obvious reasons. The other would be that blacks would have more kids, since those living in rural areas usually have more kids than city residents. As a result, the demographic situation of the country would be worse for the white minority by the time that blacks started moving to the cities for work.
So this either leads to a more bloody end to apartheid and an agrarian or to a more successful reign for the racist regime, which could continue into the present day given the circumstances elsewhere in Africa. I hope this was a satisfactory answer.