Well that is a huge question that I am probably not really qualified to answer. But if I do the best I can... Afrikaner nationalism is the easiest and most well documented one to trace. It starts to be formulated in earnest by various in particular strict Calvinists in the middle part of the 19th century. Obviously this is in tandem with the arrival of the British in the Cape Province and the resentment that a lot (but not all) of the Dutch had towards this - in particular the way the British unkindly forced them to free their slaves and stuff like that. It then develops a big mythological push with events like the great treks; later the Boer war which continues to be intellectualised by organisations like the National party and the Broederbond in the early 20th century.
There is often mention of an alleged divide between "Boers" (ie descendents of those who left the Cape) and Afrikaners (those who stayed, were more loyal to the Brits etc...) that does occasionally get mentioned these days. It was a bigger deal in terms of the Nats' mid-20th century internal divisions, broadly speaking the more radical Verwoerdian Nationalism was always more associated in particular with the Transaval ie "Boer" NP, with the Cape Dutch or "Afrikaners" being comparitively less... out there... with regards to how nasty to be to non-whites. I wouldn't overplay it's role these days though. The defining characteristic of 21st century Afrikanerdom is a massive identity crisis over "what they did", how they are supposed to fit in to modern South Africa and all the rest.
As for how their identity merged into South African national identity. Well, it's an ongoing process - but in some respects almost functions the other way round. In so far as they had to be convinced that other peoples, first the English, then the Coloureds and Africans and Indians all got to be part of "their" South African identity.
In contrast, English South African identity is much weaker. In so far as "English" South Africans are a mix of different backgrounds, mostly British, but lots of Jewish, Eastern European, Portuguese - where the "English" term just became a portmanteau for everyone who didn't speak Afrikaans and adopted english as a lingua franca. That is especially the case in the Witwatersrand; in contrast Natal had a very (very) British identity well into the second half of the 20th century. It actually toyed very seriously with trying to seceed after the Republic referendum, and continued to have the epithet "the last outpost of the British empire" for its Anglophilia for a long while after that. Eventually though, it was things like the Republic, the sense of "white" South African being back to the wall against universal condemnation during the apartheid era that mostly brought them into the fold with the Afrikaners.
As an English Canadian who grew up in Quebec - reading this I find striking parallels between the Afrikaners and the Quebecois (francophone Quebecers) and English Quebecers and anglo South Africans
