Since the English and Dutch cultures are relatively open minded and free of prejudice, how did apartheid gain such a foothold in South Africa?

Who says they were? Those cultures have a long history of slave capturing and trading, plus ruthless and cruel—even murderous—exploitation of peoples in their occupations, trading hegemonies, and colonies! Sure, after the 20th century’s World War II, with, for example, the UK’s complete withdrawal (some call it abandonment, from the way they went about it starting in the previous century), they’ve had more convincing bragging rights about their ‘liberalism’. But that’s pretty darn recent.

Because South Africa isn’t England or the Netherlands, no matter the ancestry of its white population, and has its own culture and issues.

Not to excuse them, but they were conquerors and, at least at the beginning, must have faced opposition. It is one thing to allow minorities to assimilate, and another to make your culture the dominant one where you are the minority.