It is possible that in the UK, "taken" with a meal is possibly more common than in the US. However, in my experience, these days it is really only used with "tea", as in "taken tea" or "take tea". Even then, it sounds quite formal and to me a little "stuffy", as in only used by people who want to sound "posh". "Posh" is used to describe something typical of "upper class" individuals in the UK. To "take breakfast" or "take dinner" sounds quite odd to me, but I can imagine a rather "stuffy" person saying it, as in "Shall we take breakfast together?"
I think that "After I eat breakfast, I leave for work" , is a little unnatural because "I" is used twice. I prefer "After eating breakfast, I leave for work". You could also say "Having eaten breakfast, I leave for work."
For your second example, "After achieving a goal, children should be rewarded by their parents." sounds better to me. You could also say "Having achieved a goal, children should be rewarded by their parents."