Is it possible to move to a situation where virtually all the whites in South Africa think of themselves as a single pro-Empire English-speaking people, like in Australia?
I doubt strongly that you could get an Australian analogue, if just for the reason that most white Australians were (till after WW2) descendants of immigrants from the various British nations many of which continued to see themselves as British for some time. As opposed to the Afrikaner's mixed origins of Dutch, German, French, random British or other groups (including non whites in some cases) who were never principally British.
That being said, without obvious conflict, say with an 1850s POD, perhaps the two cultures would not be so alienated. Without the attempted coup or the two wars the relationship would be far closer. Perhaps without the second war anyway, the nationalist identification with the formalisation of the Afrikaner language would not be so complete. But that being said, most cultures don't like surrendering their languages, even if to friends