I am not a native-speaker, but I have read that Americans can use either the present perfect or the past simple for recent actions. So it's right to say the following:
1- Their kids have not had or didn't have a good upbringing. We can also say that their kids have not been brought up well or were not brought up well.
2- As for the second sentence, there is nothing wrong with it. It's grammatically correct because the phrase "brought up" has been used as an adjective, not as a verb (look up well brought up in CAMBRIDGE DICTIONARY). However, we can also say that their kids were brought up badly or have been brought up badly.